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date: 18 January 2020

# Introduction

As suggested by the title, this article attempts to provide an overview of the research that highlights the macroeconomic aspects of the housing market.1 It is incomplete because the literature has been constantly growing, especially following the global financial crisis (GFC), and hence is almost impossible to review completely. Therefore, some insightful papers may not have been discussed in this article, not because they are unimportant, but simply because they have been discussed in detail elsewhere. For instance, there have been several similar attempts to discuss the literature in special issues of academic journals, survey papers, and monographs.2

According to Leung (2004), housing was not included in traditional macroeconomics. However, thanks to a few pathbreaking papers, much has changed.3 While prior literature focuses on residential investment and how it interacts with other macroeconomic variables such as business investment, gross domestic product (GDP), and so on, extant literature has quickly expanded to cover house price, transaction volume, vacancy, and so on.4 As one of the objectives of this article is to provide a general overview of the related research, we re-examine some “stylized facts” of how the macroeconomic variables (MV) on the one hand, and housing market variables (HMV), on the other hand, are correlated. It is hoped that this will inspire new research on macro-housing. Again, for the sake of inspiring new research, we then selectively review some of the literature. In the final section, we conclude this article.

# Stylized Facts

We first focus on the business cycle frequency. To establish some “stylized facts” on the “macro aspects of housing,” we follow the approach of Cooley and Prescott (1995), which provides some (unconditional) correlations between MV and HMV in the United States, where most macroeconomic research is done.5 Clearly, studies on the macroeconomic aspects of housing have used data from different countries.6 This article employs data from the United States because (a) they are the most accessible, and (b) most “macroeconomic research” uses U.S. data and hence our use of the same data will facilitate comparison. Our list of HMV includes not only the price index (which is the focus of most research), but also the number of new houses sold, the vacancy rate, and the residential investment. Owing to the well-known debate on the potential bias with respect to appraisals, we use the transaction-based house price index.7 Hence, this article trades off the length of the time-series data for a less controversial interpretation of the results. Except for a few variables such as the consumer price index (CPI) and federal funds rate (FFR), all of the other variables are represented in real terms.

Following Burns and Mitchell (1946) and most subsequent research on business cycles, this section focuses on the “cyclical components,” that is, components with periodicity between 6 and 32 quarters. As many of the variables are non-stationary, moments (including correlations) are not well defined. The variables need to be detrended, and we therefore use the following procedures. First, this article uses the band-pass filter developed by Christiano and Fitzgerald (2003) to extract the “cyclical components.”8 Second, this article not only reports those correlations, but also tests whether they are statistically significant or not. Third, this article compares two sampling periods: (a) from 1991 to 2006, which will be referred to as the pre-crisis sub-sample (PCSS); and (b) from 1991 to 2017Q3, which will be referred to as the full sample (FS). Some authors argue that there has been a “structural change” following the GFC. Therefore, the computations and comparison of the correlations in both sampling periods would reflect whether there are indeed some important differences. There is one technical issue here. Forbes and Rigobon (2002) demonstrate that if the volatility of a variable increases over time, the measured correlation could be biased. This article mitigates this concern by following Stock and Watson (2002) to first standardize the volatility of all variables, and then compute the correlations among different variables. The details are explained in the Appendix.

Table 1 Panel A shows the correlations (contemporaneous, one-period-lead, and one-period-lag) among the housing market variables and the more conventional macroeconomic variables. Table 1 Panel B shows the counterparts among the housing market variables and the macro-finance variables. The idea is to investigate whether some macroeconomic variables may be “leading” the housing market variables, or vice versa.9

Table 1. Correlation between the Cyclical Component of Housing and Macro-Variables

Housing variables

Real “purchase-only” house price index

New house sold

Vacancy rate of for-sale housing

Real private residential fixed investment

at time t Macro-variables

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

Real GDP

t-1

0.77***

0.57***

0.24*

0.14

-0.39***

-0.03

0.65***

0.54***

t

0.78***

0.66***

0.42***

0.34***

-0.44***

-0.10

0.76***

0.68***

t+1

0.74***

0.69***

0.58***

0.51***

-0.52***

-0.16

0.83***

0.76***

Unemployment rate

t-1

-0.67***

-0.27***

0.18

0.27***

-0.12

-0.28***

-0.29**

-0.12

t

-0.76***

-0.43***

-0.01

0.09

0.08

-0.20**

-0.48***

-0.32***

t+1

-0.81***

-0.59***

-0.21

-0.13

0.33***

-0.09

-0.64***

-0.51***

CPI

t-1

0.18

-0.12

-0.47***

-0.37***

0.39***

0.19**

-0.22*

-0.21**

t

0.30**

0.00

-0.33***

-0.24**

0.14

0.13

-0.05

-0.06

t+1

0.46***

0.19*

-0.18

-0.10

-0.07

0.06

0.13

0.11

Real private nonresidential fixed investment

t-1

0.71***

0.31***

-0.04

-0.22**

-0.03

0.18*

0.39***

0.15

t

0.78***

0.45***

0.14

-0.04

-0.18

0.12

0.55***

0.33***

t+1

0.81***

0.58***

0.31**

0.16

-0.38***

0.03

0.69***

0.51***

Real personal consumption expenditures

t-1

0.66***

0.41***

0.02

0.06

-0.25**

0.02

0.45***

0.41***

t

0.72***

0.52***

0.20

0.22***

-0.33***

-0.03

0.60***

0.55***

t+1

0.73***

0.58***

0.37***

0.37***

-0.43***

-0.08

0.70***

0.65***

t-1

-0.77***

-0.30***

-0.17

-0.10

0.25**

0.00

-0.58***

-0.45***

t

-0.78***

-0.39***

-0.34***

-0.27***

0.47***

0.11

-0.72***

-0.59***

t+1

-0.77***

-0.48***

-0.50***

-0.44***

0.69***

0.23**

-0.82***

-0.69***

Housing variables

Real “purchase-only” house price index

New house sold

Vacancy rate of for-sale housing

Real private residential fixed investment

at time t Macro-variables

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

Nominal federal funds rate

t-1

0.54***

0.56***

0.02

0.06

0.15

0.23**

0.27**

0.33***

t

0.61***

0.64***

0.08

0.18*

0.01

0.08

0.38***

0.47***

t+1

0.64***

0.69***

0.17

0.29***

-0.19

-0.07

0.50***

0.59***

Real federal funds rate

t-1

0.48***

0.63***

-0.28**

-0.06

0.35***

0.23**

0.06

0.13

t

0.62***

0.65***

-0.16

0.01

0.22*

0.08

0.24*

0.22**

t+1

0.66***

0.55***

-0.03

0.06

0.01

-0.03

0.39***

0.30***

t-1

-0.64***

-0.62***

-0.10

-0.20**

-0.08

-0.07

-0.36***

-0.44***

t

-0.69***

-0.64***

-0.19

-0.32***

0.15

0.11

-0.46***

-0.55***

t+1

-0.68***

-0.61***

-0.27**

-0.40***

0.42***

0.28***

-0.55***

-0.65***

t-1

-0.11

-0.43***

-0.39***

-0.45***

-0.08

-0.01

-0.22*

-0.37***

t

-0.11

-0.36***

-0.33***

-0.43***

-0.15

0.11

-0.19

-0.28***

t+1

-0.03

-0.21**

-0.28

-0.35***

-0.12

0.21**

-0.14

-0.17*

t-1

-0.38***

-0.42***

-0.33***

-0.29***

0.34***

-0.06

-0.48***

-0.45***

t

-0.30**

-0.52***

-0.37***

-0.41***

0.47***

0.12

-0.49***

-0.49***

t+1

-0.21

-0.57***

-0.44***

-0.50***

0.65***

0.31***

-0.49***

-0.49***

S&P 500 index (real terms)

t-1

0.74***

0.57***

0.43***

0.22**

-0.49***

-0.04

0.73***

0.55***

t

0.70***

0.68***

0.55***

0.41***

-0.80***

-0.12

0.78***

0.66***

t+1

0.62***

0.74***

0.64***

0.55***

-0.53***

-0.20**

0.78***

0.72***

Several observations are in order. First, most correlations are significant, suggesting that the (aggregate) housing market variables and the macroeconomic variables are indeed closely related. In other words, the “macro-housing” literature is justified. Second, while most correlations in the two sampling periods are similar, there are cases with some noticeable differences. For instance, in the PCSS, the real GDP and vacancy rate of the for-sale housing were significantly and negatively correlated. The unemployment rate and house prices were also significantly and negatively correlated. The idea is simple. When the economy receives a positive shock, firms hire more labor and people are more willing to purchase housing units, resulting in an increase in the aggregate output and a decrease in the unemployment rate. At the same time, the house prices increase and the vacancy rate of the for-sale housing market decreases. However, for the full sample, such intuitive correlations disappear. Clearly, these are simple correlations and they provide no proof of causality. Nevertheless, they suggest perhaps that there is a change in the dynamics between the macroeconomy and the housing market, which may be related to the “jobless recovery” following the GFC in the sense that the aggregate GDP rebounds much faster than the labor market. Moreover, the slow recovery of the labor market also affects the recovery of the housing market.10

Another set of important correlations is related to the level of the CPI. This is clearly important; many undergraduate students were taught that the CPI and inflation are at the core of macroeconomics.11 Moreover, the mandate of several central banks is tied to some sort of “price stability.” Therefore, we believe that it is important to examine the correlations between the CPI on the one hand, and the housing variables on the other hand. Table 1 Panel A shows that before the GFC, the CPI was positively correlated to house prices. However, the statistical significance disappears when we consider the full sample, suggesting that the relationship between the CPI and house prices has also “weakened” after the GFC. Moreover, the correlations between the CPI on the one hand and other housing variables on the other hand also tend to be insignificant, which is consistent with the recent literature that suggests that the relationship between the inflation rate and other macroeconomic variables may have changed after the GFC.12

The fourth row of Table 1 Panel A also shows that the real private non-residential fixed investment had a strong positive correlation with real house price, and a negative correlation with the for-sale housing vacancy rate in the PCSS. This intuition is simple to explain. When the economy is booming, more investment is made for non-residential fixed assets. At the same time, the demand for housing increases and hence house prices increase and the vacancy rate decreases, which explains the observed correlations. Conversely, once we include the post-crisis years, the correlation between non-residential fixed investment and house prices weakens, and the correlation between the non-residential fixed investment and the for-sale housing vacancy rate becomes insignificant. Similarly, while non-residential fixed investment and residential investment are evidently correlated,13 the correlations seem to weaken after the GFC. In fact, the correlation between the prior period of non-residential investment and the current period of residential investment was around 0.4 before the GFC. The same correlation drops to about 0.15 and becomes statistically insignificant in the full sample. Again, we observe a “weakening” of the link between the macroeconomy and the housing market.

The same “weakening” may also be exhibited in consumption. The “wealth effect of consumption” is being discussed even in undergraduate macroeconomic textbooks. There is extensive literature on whether (and how) fluctuations in housing wealth and stock market wealth affect consumption.14 Table 1 Panel A displays many correlations between the aggregate private consumption on the one hand, and the housing market variables on the other hand. The correlations between consumption and house price ranged from 0.66 to 0.73 before the GFC. However, when we consider the full sample, those correlations drop to between 0.41 and 0.58. The intuitive correlations between consumption and the vacancy rate are negative and statistically significant. When the economy booms, people consume more and are simultaneously more willing to buy a house, and hence the vacancy rate drops. However, those correlations become insignificant when we consider the full sample. Again, it only adds to the evidence that the correlations between housing market variables and macroeconomic variables weakened after the GFC.

Similarly, before the GFC, the correlations between the trade surplus and house prices were high, and were around -0.77.15 At the same time, the trade surplus is positively related to the vacancy rate of for-sale housing. The intuition here is also simple to explain. When the trade account has a surplus, the capital account is in deficit, suggesting that there is an outflow of capital. With an outflow of capital, it is natural to expect that house prices fall and the vacancy rate of housing increases.16 However, when the years following the GFC are included, the correlations between trade surplus and house prices drop to between -0.30 and -0.48. The correlations between the trade surplus and the vacancy rate of for-sale housing are also significantly reduced, and some even become statistically insignificant.

Table 1 Panel B shows the correlations between some “macro-finance” variables on the one hand, and the housing market variables on the other hand. We begin with the FFR, which is often considered as the “policy rate,” an indicator of the “tightness” of the monetary policy. The first row presents the results for the nominal FFR and the second row presents the counterpart for the real FFR, which is defined as the difference between the nominal FFR and the inflation rate (based on the CPI). Note that after the GFC, the Federal Reserve cuts the nominal interest rate to almost zero, and many authors argue that the “zero lower bound” (ZLB) introduces distortions in both parameter estimation as well as welfare cost. To formally study the ZLB, however, is a very involved process.17 Therefore, we introduce the real FFR instead, which does not have a ZLB for comparison. Comparing the first and second rows of Table 1 Panel B, it is evident that the results related to the nominal and real FFR are usually similar, which is perhaps because the CPI inflation rate is dominated by high-frequency components. As the Christiano and Fitzgerald band-pass filter accurately extracts only the business cycle components (i.e., components with periodicity between 6 and 32 quarters), the impact of inflation on the FFR is minimal. Clearly, the FFR is positively correlated to the real house price, and only weakly correlated with other housing market variables. This may sound counter-intuitive, because a higher interest rate should discourage people from investing. However, if we take the endogeneity of the FFR into consideration, the correlation between the FFR and house prices can be understood better. Nakamura and Steinsson (2017) explain this well:18

The Federal Reserve lowered interest rates aggressively in 2008 as evidence mounted that the economy was heading for a severe downturn. Suppose one sought to use this variation in monetary policy to estimate the effect of monetary policy on the economy by estimating an OLS regression of the change in output on the change in interest rates. Doing this might lead one to conclude that reductions in interest rates lead to decreases in output. Would this constitute convincing evidence on the effects of monetary policy? Of course not. The reason the Fed was lowering interest rates was that other factors—such as rapidly falling home prices and their effects on the balance sheets of households, firms, and banks—were negatively affecting the economy. In a simple OLS regression, these other factors would confound the effects of the change in monetary policy. This approach would, therefore, uncover not the pure effect of the interest rate reduction, but rather, the combined effect of the interest rate reduction and the adverse macroeconomic factors that led the Fed to undertake it.

In the literature, therefore, researchers have used other interest rates as proxies of the credit market conditions, and examined the co-movements of those interest rates and the housing market variables.19 Table 1 Panel B also shows how these interest rates are related to housing market variables. For instance, clearly the term spread, which is the difference between the long-term and short-term interest rate, is negatively correlated with house prices and residential investment. Interestingly, while the correlations between the macroeconomic variables on the one hand and the housing market variables on the other hand weakened after the GFC, the correlation between the term spread and the residential investment became stronger. Furthermore, the correlations between the term spread and the number of new houses sold are either enforced or turning from statistically insignificant to negatively significant.

The Treasury–Eurodollar rate (TED) spread is the difference between a measure of the interbank interest rate and the risk-free rate. An increase in TED is often interpreted as an increase in the cost of external funds for financial intermediaries. Since interbank loans tend to be relatively short-lived, usually less than a month, or even overnight, while mortgages or real estate development often take more than a year, the latter would not be financed by interbank loans. Hence, it may not be surprising that the TED’s correlations with either the house price or the residential investment before the GFC are weak. However, when we examine the same correlation for the full sample, the TED spread is negatively correlated with both house prices and the residential investment. It seems to be “leading” both house prices and residential investment. The correlations between the TED spread and the number of new houses sold are also strengthened after the GFC. Such changes may be related to two factors: (a) the manner in which the Federal Reserve Bank implements the monetary policy,20 and (b) the adoption of the unconventional monetary policy (UMP) since the GFC. Recent research confirms that UMP impacts financial institutions and the economy through different channels.21 Table 1 Panel B simply confirms that research in the context of the housing market.

The case for the external finance premium (EFP) is interesting. The EFP is the difference between the prime bank lending rate and the three-month Treasury bill (T-bill) rate. An increase in EFP is often interpreted as an increase in the cost of external funds for non-financial firms, and it has been used in many types of research.22 Table 1 Panel B shows that its correlation with the house price is negative. On the other hand, it had a positive and significant correlation with the vacancy rate before the GFC. The idea is that when the credit market is tight, it is difficult for both firms and households to obtain credit. And when families cannot get credit, many of them cannot purchase homes. Hence, both the EFP and the vacancy rate of the for-sale housing increase, and the house prices decrease. When we consider the full sample, that is, when the GFC and the subsequent years are taken into consideration, the correlations between the EFP and the vacancy rate of for-sale housing weaken, and some even become statistically insignificant. At the same time, the negative correlation between the EFP and house prices is “strengthened” in the sense that it becomes more negative. Notice that several changes happen in the post-crisis period. It is instructive to perceive that the GFC hits the economy as if some wealth is suddenly being “destroyed,” or as if the economic agents realize that some asset is overvalued (Brunnermeier & Sannikov, 2014). Economic agents rebalance their portfolios. To some people, this means trading down their homes or selling their seasonal homes. To others, it means defaulting on mortgage loans—thus, the housing supply increases and the equilibrium house price drops. At the same time, the rebalancing leads to less supply in credit. Firms find it more difficult to borrow. While the EFP was usually no more than 1.5% (seasonally adjusted) before 2008, it reached as high as 3% during 2008–2009 period. Different policy measures were imposed by the U.S. government to stabilize the economy (Bernanke, 2017). As a result, the EFP returns to normal, and the house price gradually hits bottom and then rebounds.

On the other hand, the legal aspects of the foreclosure process can be time-consuming. It also takes time for economic agents to accumulate enough wealth to purchase homes under more stringent mortgage loan application rules. The vacancy rate returns to pre-crisis levels at a much slower pace.

There is extensive literature on the co-movement between the housing market and the stock market.23 Table 1 Panel B shows that the correlations between the house price and stock price (measured by the S&P 500 index) are indeed positive and statistically significant, both before and after the GFC. There is a subtle difference here. In the PCSS, the correlation between the lagged stock price of one period and the house prices of the current period is larger than the contemporaneous correlation, which in turn is larger than the correlation between the stock price of one period ahead and the house prices of the current period, suggesting that the stock price is somewhat leading the house price. As we consider the full sample, that is, including the years after the GFC, we find the reverse pattern, suggesting that the stock price is somewhat lagging the house price. Clearly, a formal testing for leading versus a lagging relationship would need more sophisticated tests and perhaps a longer time series. Nevertheless, this suggests a possible change in the relationship between the stock and housing markets. At the same time, comparing the correlations between the stock price on the one hand, and the other HMV on the other hand (including the number of new houses sold, the vacancy rate of for-sale housing, and the private residential fixed investment), a “general weakening” of the relationship emerges. For instance, the relationship between the stock price and the number of new houses sold remains positive, but the numerical values drop in the full sample. Some correlations between the stock price and the housing vacancy rate turn from negative and significant to insignificant. The correlations between the stock price and the residential investment remain positive but are numerically much smaller. Clearly, it would be premature to conclude a “de-coupling” between the stock market and housing market. At the same time, it seems important to examine whether and how the relationship between the stock market and housing market has changed after the GFC.

## Medium Cycle Frequency

Thus far, we have focused on business cycle frequency, which many macroeconomic studies have also focused on. However, some authors propose that we should focus on the “longer run.”24 Here, we follow Comin and Gertler (2006) to study components with periodicity between 32 and 80 quarters, which we label as “medium-run components.” We are aware that our sampling period is relatively short and hence the estimate of the medium cycles may not be as accurate as it should be. Constrained by the unavailability of data, we can only recognize the limitations and proceed nevertheless. To facilitate the comparison, we repeat the exercise in the previous section with the “medium-run components.” The results are reported in Table 2, which are identical to those in Table 1, except that the correlations are now correlations among medium-run components rather than cyclical components. As in the case of cyclical components, most correlations are statistically significant, suggesting that the macroeconomy and the housing market are closely related in the medium run as well. At the same time, we do observe some differences between the PCSS and the full sample, and focus our discussion on those differences.

Table 2. Correlation between the Medium-Cycle Component of Housing and Macro-Variables

Housing variables

Real “purchase-only” house price index

New house sold

Vacancy rate of for-sale housing

Real private residential fixed investment

at time t Macro-variables

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

Real GDP

t-1

0.57***

0.81***

0.61***

0.73***

-0.11

-0.14

0.62***

0.80***

t

0.58***

0.79***

0.65***

0.79***

-0.16

-0.24***

0.65***

0.83***

t+1

0.56***

0.75***

0.68***

0.83***

-0.26**

-0.34***

0.67***

0.85***

Unemployment rate

t-1

-0.34***

-0.69***

-0.35***

-0.67***

-0.12

0.14

-0.38***

-0.72***

t

-0.36***

-0.67***

-0.41***

-0.72***

-0.07

0.24***

-0.43***

-0.76***

t+1

-0.35***

-0.63***

-0.45***

-0.76***

0.01

0.33***

-0.46***

-0.77***

CPI

t-1

-0.57***

-0.29***

-0.53***

-0.24***

0.56***

0.25***

-0.49***

-0.20**

t

-0.53***

-0.26***

-0.49***

-0.20**

0.55***

0.22**

-0.44***

-0.15

t+1

-0.50***

-0.23**

-0.46***

-0.17*

0.58***

0.22**

-0.40***

-0.13

Real private nonresidential fixed investment

t-1

-0.32***

0.21**

-0.25**

0.26***

0.26**

-0.05

-0.27**

0.28***

t

-0.30**

0.20**

-0.23*

0.29***

0.28**

-0.09

-0.24*

0.31***

t+1

-0.30**

0.18*

-0.21*

0.32***

0.30**

-0.13

-0.21*

0.33***

Real personal consumption expenditures

t-1

0.72***

0.86***

0.80***

0.78***

-0.38***

-0.19**

0.78***

0.83***

t

0.70***

0.84***

0.81***

0.83***

-0.42***

-0.29***

0.78***

0.86***

t+1

0.67***

0.80***

0.81***

0.86***

-0.51***

-0.38***

0.77***

0.88***

t-1

-0.99***

-0.98***

-0.94***

-0.84***

0.39***

0.17*

-0.96***

-0.90***

t

-0.97***

-0.96***

-0.95***

-0.88***

0.46***

0.28***

-0.96***

-0.92***

t+1

-0.95***

-0.93***

-0.96***

-0.92***

0.59***

0.38***

-0.95***

-0.93***

Housing variables

Real “purchase-only” house price index

New house sold

Vacancy rate of for-sale housing

Real private residential fixed investment

at time t Macro-variables

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

1991Q1–2006Q4

1991Q1–2017Q3

Nominal federal funds rate

t-1

-0.26**

0.17*

-0.49***

-0.19**

0.93***

0.70***

-0.34***

-0.03

t

-0.14

0.25***

-0.37***

-0.08

0.89***

0.64***

-0.22*

0.08

t+1

-0.05

0.33***

-0.28**

0.02

0.84***

0.57***

-0.12

0.18*

Real federal funds rate

t-1

-0.47***

0

-0.57***

-0.27***

0.78***

0.63***

-0.48***

-0.15

t

-0.38***

0.07

-0.48***

-0.17*

0.78***

0.59***

-0.38***

-0.05

t+1

-0.32***

0.14

-0.41***

-0.08

0.77***

0.54***

-0.31**

0.04

t-1

0.27**

-0.18*

0.31***

0

-0.60***

-0.42***

0.25**

-0.10

t

0.20

-0.22**

0.23*

-0.09

-0.58***

-0.35***

0.17

-0.18*

t+1

0.17

-0.25***

0.17

-0.16*

-0.55***

-0.30***

0.10

-0.25***

t-1

-0.45***

-0.02

-0.59***

-0.43***

0.75***

0.80***

-0.53***

-0.30***

t

-0.35***

0.08

-0.51***

-0.32***

0.79***

0.78***

-0.42***

-0.19*

t+1

-0.31**

0.16*

-0.43***

-0.22**

0.79***

0.75***

-0.34***

-0.08

t-1

-0.39***

-0.48***

-0.58***

-0.68***

0.30**

0.38***

-0.54***

-0.65***

t

-0.35***

-0.43***

-0.58***

-0.68***

0.36***

0.45***

-0.53***

-0.63***

t+1

-0.32***

-0.38***

-0.58***

-0.67***

0.44***

0.52***

-0.52***

-0.61***

S&P 500 index (real terms)

t-1

-0.36***

0.12

-0.23*

0.25***

-0.03

-0.25***

-0.31***

0.23**

t

-0.39***

0.09

-0.25**

0.26***

0.03

-0.26***

-0.32***

0.23**

t+1

-0.42***

0.05

-0.27**

0.25***

0.09

-0.27***

-0.33***

0.22**

Notes: Our definition of “medium cycles” follows Comin and Gertler (2006), which captures the components with periodicity between 32 and 80 quarters.

Given the structure, we begin with the first row in Table 2 Panel A, the correlations between the GDP on the one hand and housing market variables on the other hand. Interestingly, while the correlations between GDP and housing market variables are weakened in the business cycle frequency, they are strengthened in the medium-cycle frequency. For instance, the correlations between the cyclical components of GDP and the vacancy rate of for-sale housing turn from negative and significant to insignificant, while the medium-cycle counterparts turn from mostly insignificant to mostly negative and significant. Undoubtedly, this is a challenge for future research to construct a theoretical model to explain these changes.25

Such a challenge is not limited to the relationship between GDP and housing market variables. As we study the second row of Table 2 Panel A, we find that the correlations between the medium-run components of the unemployment rate and house price are strengthened, while the counterparts in cyclical components are weakened. Moreover, while the correlations between the cyclical component of the unemployment rate and the number of new houses sold are mostly insignificant, they are negative and significant in the medium-run component. Furthermore, the correlations are strengthened after the GFC. While the correlations between the cyclical components of the unemployment rate and the vacancy rate of for-sale housing are mostly negative in the full sample, the counterparts in the medium-run components are mostly positive in the full sample, which seems to be more intuitive. When the economy is hit by a bad shock, the unemployment rate increases and hence the demand for housing drops, which in turn leads to an increase in the vacancy rate of housing.

Note that there are also situations in which patterns in cyclical components and medium-run components are consistent. For instance, the third row of Table 2 Panel A shows that the correlations between the medium-run components of CPI and house price are negative and significant, but the correlations are obviously weakened in the full sample. This is in line with the finding in the cyclical component that the positive correlation between the two variables turn from positive and significant to mostly insignificant. In fact, the correlations between the medium-run components of CPI and other housing market variables (the number of new houses sold, a vacancy rate of for-sale housing, and private residential fixed investment) are dramatically weakened in the full sample. Thus, it is in line with the idea that changes in nominal variables have limited long-run impact on real variables.26

The case for private non-residential fixed investment also poses a challenge. It is well known that the non-residential fixed investment and the residential counterparts are positively correlated with business cycle frequency.27 Table 2 Panel A reveals that, in fact, the counterpart in medium-cycle frequency is negatively correlated before the GFC. It, however, turns out to be positive and significant in the full sample. Similarly, the correlations between the medium-run components of the non-residential investment and house price are negative before the GFC and become positive and significant in the full sample. This is in contrast to the case of cyclical components, which remain positive and significant in the PCSS and the full sample.

The correlations between the medium-run components of trade surplus and housing prices somewhat differ from the cyclical component counterpart. In the case of cyclical components, the correlations between the trade surplus are reduced from the -0.76 to -0.78 range to the -0.30 to -0.48 range as we extend from the PCSS to the full sample. In the case of medium-run components, however, the correlations are always in between -0.90 and -0.99, suggesting that in real terms, the international capital flow and house prices are more correlated in the medium run than in the business cycle frequency. In fact, the correlations between the trade surplus on the one hand, and the number of new houses sold and the private residential fixed investment on the other hand, display a similar degree of co-movements. Thus, even for a large country such as the United States, where net exports constitute around 10% of GDP, international capital flows are important for the housing market. Future research may further explore this dimension.

Table 2 Panel B exhibits the correlations between the medium-run components of the macro-finance variables on the one hand and housing market variables on the other hand. Its format closely follows that of Table 1 Panel A, where the counterpart of the business cycle components is displayed. The first row shows the correlation between the nominal FFR and the housing market variables. Its “pattern” dramatically differs from the counterparts of the cyclical components. For instance, the correlations between the medium-run component of the nominal FFR and the house price are statistically significant, yet numerically small (in the range of 0.17 to 0.33 in the full sample). In contrast, in the case of cyclical components, the correlations are in the range of 0.55 to 0.69, and they are statistically significant. The correlations between the medium-run components of the nominal FFR and the vacancy rate of for-sale housing are in the range of 0.84 to 0.93 in the PCSS, and in the range of 0.57 to 0.70 in the full sample. Although the correlations are “weakened,” they remain statistically significant. On the other hand, the cyclical counterpart is mostly insignificant. At the same time, the correlations between the medium-run components of the nominal FFR and private residential fixed investment become marginally significant or even insignificant in the full sample. The cyclical component counterparts are also statistically significant at the 1% level and in the range of 0.33 to 0.60. In other words, the co-movements between the nominal FFR and the housing market are very different in the business cycle frequency and in the medium cycles.

In the business cycle frequency, the correlations between nominal FFR and housing market variables, and those between the real FFR and housing market variables, are similar. In the medium run, however, this is not always the case. For instance, the second row of Table 2 Panel B shows that the correlations between the real FFR and real house price are negative and significant before the GFC. When we consider the full sample, those correlations become numerically small and statistically insignificant. The counterpart in the business cycle frequency is always positive and significant, whether in the PCSS or in the full sample. In other cases, the behavior of the medium-run components of nominal and real FFR seem to be similar. Both the nominal and real FFR are positively correlated with the vacancy rate of the for-sale housing in the medium run, while almost uncorrelated with the vacancy rate in the business cycle frequency. Both the nominal and real FFR are statistically correlated with the number of new houses sold before the GFC in the medium run and then become almost uncorrelated in the full sample, while they are typically uncorrelated with the number of new houses sold in the business cycle frequency. As monetary research constitutes a significant part of the macroeconomic research, accounting for these differences may be important for future modeling development.

The third row of the Table 2 Panel B shows how the term spread correlates with the housing market variables. It is interesting to note that before the GFC, the medium-run term spread is only weakly correlated with the house price in the PCSS. It becomes statistically correlated with the house price in the full sample. Yet the numerical values are somewhat small, in the range of -0.18 to -0.25, which is in sharp contrast to the case of business cycle frequency. The correlations between the cyclical components of term spread and house price are always between -0.61 and -0.70, and they are statistically significant. The cyclical component of the term spread is also negatively correlated with the number of new houses sold in the full sample, with correlations between -0.20 and -0.40. In the medium run, however, the two variables are more or less uncorrelated.

Similarly, a dramatic difference is observed in the TED spread. The fourth row of Table 2 Panel B clearly shows that the correlations between the medium-run components of the TED spread and house price are negative and significant before the GFC and become almost insignificant in the full sample. In the cyclical component counterpart, while the two variables are not correlated before the GFC, they become negatively and significantly correlated in the full sample. Moreover, the medium-run components of the TED spread and the vacancy rate of the for-sale housing are significantly correlated. The correlations range between 0.75 and 0.80, whether in the PCSS, or during the full sample. The cyclical components of the same pair of variables are almost uncorrelated.

While the relationships between the EFP and housing market variables are similar in the business cycle frequency and in the medium-cycle frequency, the relationships between the stock price (measured by the S&P 500 index, in real terms) and housing market variables are very different in the two frequencies. Recall from Table 1 Panel B that the cyclical components of the stock price and house price are positively (and significantly) correlated, both before the GFC and in the full sample. Table 2 Panel B, however, shows that the medium-cycle components of the same pair of variables are negatively and significantly correlated before the GFC, with correlations between -0.36 and -0.42. In the full sample, the two asset prices become uncorrelated. Similarly, the cyclical components of the stock price and the number of new houses sold are positively and significantly correlated, both before the GFC and in the full sample. However, the medium-cycle components of the same pair of variables are negatively and significantly correlated before the GFC, with correlations between -0.23 and -0.27. In the full sample, the stock price and number of new houses sold become positively correlated. The correlations are statistically significant, but economically weak, in the range of 0.25 to 0.26. The cyclical components of the stock price and private residential investment are positively and significantly correlated, both before the GFC and in the full sample. The correlations are 0.55 or above. However, the medium-cycle components of the same pair of variables are negatively and significantly correlated before the GFC, with correlations between -0.31 and -0.33. In the full sample, the stock price and residential investment become positively correlated. The correlations are statistically significant, but economically weak, in the range of 0.22 to 0.23.

## A Quick Summary

Here is a quick summary. For the business cycle frequency (or “cyclical components,” i.e., components with periodicity between 6 and 32 quarters), while macroeconomic variables and housing market variables are correlated in many dimensions, some “stylized facts” indeed change after the GFC. More specifically, many intuitive correlations between MV on the one hand and HMV on the other hand are weakened. At the same time, several measures of interest rates have their correlations with HMV strengthened. These changes may be important because theoretical models have been built to explain the “old stylized facts,” and this is how our “intuitions” are formed. The revision of those stylized facts may be a call for refining some existing models of macro-housing, which it is hoped will be consistent with the “stylized facts” both for the period before the GFC as well as the period after.28 For the medium-cycle frequency, some correlations follow changes similar to their counterparts in business cycle frequency, while others are very different. Apparently, existing models are relatively silent on these differences, which reinforces the call for more serious theoretical modeling efforts.

Here are some qualifications of our results. First, all of these correlations are unconditional and bilateral. Although they are statistically significant, they do not indicate the direction of causality. It is possible that the correlations are driven by some “third factor.” While some of the unconditional correlations are insignificant, it is possible that the conditional counterpart of these correlations is significant, that is, when some third variable is being held constant. In addition, if some variables exhibit non-linear dynamics such as regime-switching or contain a “bubble” component, the unconditional correlations may not provide us with the full picture.29 Moreover, all of these correlations are based on aggregate data. Recent research, on the other hand, tends to use micro-data. In the next section, we will, therefore, review some of those works.

# A Review of Selected Literature

Although recent years have witnessed a rapid growth in the macro-housing literature, this section only provides a review of a subset of this literature. The order and topics selected are somewhat arbitrary. The only objective is to provide a (partial) review that will facilitate the exchange of ideas and perhaps encourage even more research in this area.

## Bankruptcy and Mortgage Contract

In their Mayekawa Lecture presented at the Bank of Japan, Goodhart and Tsomocos (2011) argue the following: “What is the main limitation of much macroeconomic theory, among the failings pointed out by William R. White at the 2010 Mayekawa Lectures? We argue that the main deficiency is a failure to incorporate the possibility of default, including that of banks, into the core of the analysis . . . .” Recent research efforts have indeed addressed such a concern. In particular, authors have introduced heterogeneity in banks, firms, households, and their potential defaults in macroeconomics.30 In the context of macro-housing, it is natural to consider mortgage default. It follows that one should first introduce mortgage debt in macro models. Chambers, Garriga, and Schlagenhauf (2009a, 2009b, 2009c), among others, propose overlapping generation models (OGM) with stochastic labor income and endogenous tenure choice (to rent or to own).31 They also compare different types of mortgage contracts and study their implications in equilibrium. Their models can match some stylized facts of the U.S. aggregate economy and the housing market, including the homeownership rates across different age groups. Iacoviello and Pavan (2013) study an OGM with aggregate shock and match some stylized facts of housing and mortgage debt over the business cycles.

Carlos Hatchondo, Martinez, and Sánchez (2015), among others, go one step further by building a model in which the house prices are characterized by a stochastic process and allow agents to default on the mortgage loan. This step significantly complicates the analysis. For instance, in the case of recourse mortgage, if a borrower fails to make the scheduled mortgage payment, the lender has the right to seize the other liquid assets of the borrower. Hence, the saving decision is related to the default decision and hence the dynamic optimization problem involves interacting continuous and discrete choices, and some conventional methods of modeling mortgage contract may not be suitable in this context.32 Based on their structural model, they conduct counter-factual experiments and find that combining recourse mortgage and a loan-to-value (LTV) ratio limit may mitigate the sharp increase in the default rate after a sharp decline in house prices. Corbae and Quintin (2015) study a model in which agents stochastically age over time. The house price is also exogenous. The agents can choose a high or low level of down payment on the mortgage contract. Based on their structural model, they conclude that high-leverage loans originating before the financial crisis are responsible for the high foreclosure rate following the crisis. Apparently, both Carlos Hatchondo et al. (2015) and Corbae and Quintin (2015) point to the policy recommendation that some minimum level of down payment may mitigate the severity of the crisis. Mitman (2016) also builds a dynamic model with an exogenous house price process, an endogenous tenure choice, different ways of “filing bankruptcy” (Chapter 7 or Chapter 13), and foreclosure that may or may not happen.33 He also exploits the cross-state difference in the default laws in his modeling and calibration, and subsequently examines two policies, namely, the Bankruptcy Abuse Prevention and Consumer Protection Act (BAPCPA), which was imposed in 2005, and the Home Affordable Refinance Program (HARP), which was imposed in 2009. He finds that the BAPCPA reform significantly reduces the number of bankruptcy cases, and simultaneously leads to a significant increase in foreclosures during the Great Recession. He also finds that the HARP program enables households with high LTV to refinance their mortgages, thereby leading to significant welfare gains to those households.34

Justiniano, Primiceri, and Tambalotti (2013, 2015a, 2015b, 2016), among others, argue that it may be important to distinguish borrowing constraint from lending constraint. In their models, borrowing constraint is the same as the collateral constraint imposed by Kiyotaki and Moore (1997), as the value of debt cannot exceed the product of an exogenously imposed parameter $θ$ and the total value of the collateral. In this case, a “loosening” of a borrowing standard refers to an increase in the value of $θ$, capturing the idea that the same value of collateral can now “support” a larger amount of borrowing at the individual level. The lending constraint, on the other hand, refers to an upper bound on the total amount of mortgage lending that households can obtain. Justiniano et al. (2015a) show in their appendix B that the lending constraint is equivalent to a leverage constraint faced by financial intermediaries. They find that it is indeed the lending constraint, rather than the widely used borrowing constraint, that is more consistent with the four stylized facts related to the Great Recession: (a) house prices rise dramatically between 2000 and 2006, (b) household mortgage debt also increases dramatically during the same period, (c) mortgage debt and house price increase in parallel, and (d) real mortgage interest rates decline during that period. They illustrate their mechanism with two types of agents and an endogenous house price.35 However, that series of models does not consider mortgage default and household bankruptcy.

If both the default and endogenous house price dynamics are important for our understanding of the housing market, it is natural to conjecture that combining both features will produce a very robust analysis.36 It should be noted that even with the exogenous house price process, models with default and bankruptcy are typically difficult to solve. Thus, introducing endogenous house prices makes the task even more difficult. There are a few papers that have taken up this challenge. For instance, Favilukis, Ludvigson, and Van Nieuwerburgh (2017) consider an overlapping generation model in which agents are subject to both idiosyncratic and aggregate risks on the one hand and have limited access to instruments that can diversify those risks on the other hand. Thus, a distribution of (ex post) heterogeneous households will be generated even if the agents were identical ex ante. However, the agents are also heterogeneous ex ante. In Favilukis et al.’s (2017) model, a small fraction of agents receives a bequest from their parents and makes a bequest to their offspring. At the same time, most agents start with little wealth when they start to work, which creates a non-trivial wealth distribution in the model. In addition, there are participation costs for the equity market, transaction costs for the housing market, and borrowing costs, which have important implications for asset markets. For instance, when the economy is hit by an adverse aggregate shock, some agents who have (relatively) lower levels of income and wealth may choose not to participate in asset markets, leaving the relatively rich to actively participate in these markets.37 If the aggregate shock turns favorable, then those who held assets during the downturn capture a bigger share of the capital gains. In other words, with positive asset market participation costs, the distributional dynamics of income and wealth and the dynamics of asset prices can interact in a non-trivial manner. In fact, the model can then simultaneously match some macro-stylized facts, life-cycle age–income profiles, and asset returns reasonably well.

While credit market conditions and beliefs are often cited as the driving forces behind the Great Recession, their relative importance is still not known. Kaplan, Mitman, and Violante (2017) address this question by building a dynamic general equilibrium model with three different types of aggregate shocks. The first type of shock is the traditional shock on aggregate labor productivity. The second type of shock, which is significant, impacts a group of parameters that capture credit market conditions, including the maximum LTV ratio at mortgage origination, the maximum payment-to-income (PTI) level, mortgage origination cost, and so on. The third type of shock is related to a clever formulation of the aggregate preference for housing $ϕ$. In their model, $ϕ$ evolves stochastically and follows a three-state Markov process, $ϕ∈{ϕL*,ϕL,ϕH}$, where $ϕL*=ϕL<ϕH$. Thus, the preferences for housing in the periods are the same when the economy is in the state of $ϕL*,ϕL$. On the other hand, the two states differ in terms of the (conditional) probability of transferring to the state of $ϕH$, where a higher preference weight is put on housing. Kaplan et al. (2017) therefore distinguish the news shock or belief shock (which is a transition between $ϕL*$ and $ϕL$) from an actual preference shock (which is a transition between $ϕL*$ or $ϕL$ and $ϕH$).

There are other important assumptions in Kaplan et al. (2017). For instance, they assume that renters cannot borrow as they do not have collateral. However, homeowners can use their houses as collateral and borrow through a home equity line of credit (HELOC). HELOC is refinanced on a period-by-period basis and hence the amount of borrowing is affected by the time-varying house value. It is clearly very different from a mortgage loan, where the amount of borrowing is affected by the house value during the origination period only. They carefully calibrate their model to match both cross-sectional and time-series facts related to the housing market.38 In addition, they perform a series of counterfactual experiments to disentangle different effects, including: (a) households’ belief in housing demand remains fixed over time; (b) households believe that they themselves will increase the demand for housing in the future, and at the same time believe that others will not; (c) mortgage lenders believe that households will increase their demand for housing in the future, while rental firms and households do not; (d) mortgage lenders are pessimistic about the chance of households increasing their housing demands, while the rental firms and households are in fact subject to the belief shock, and so on. They find that it is indeed important to include the rental sector, and it is the belief shock rather than taste shock (the actual change in preference parameters) that accounts for the observed boom–bust dynamics. Contrary to some earlier studies, Kaplan et al. (2017) find that the relaxation and subsequent tightening of the credit market conditions are relatively minor in explaining the observed boom and bust in consumption and house prices. In Kaplan et al. (2017), the belief change is exogenous. The authors study endogenous belief dynamics in an environment with informational frictions. We will discuss some of their contributions in the section “Search and Belief.”

## Search and Belief

The application of the search-and-bargaining theory in housing market research has been widely recognized. For instance, according to the Nobel Committee (2010), “This year’s three Laureates have formulated a theoretical framework for search markets. Peter Diamond has analyzed the foundations of search markets. Dale Mortensen and Christopher Pissarides have expanded the theory and have applied it to the labor market.. . . Search theory has been applied to many other areas in addition to the labor market. This includes, in particular, the housing market. The number of homes for sale varies over time, as does the time it takes for a house to find a buyer and the parties to agree on the price.” Clearly, it is impossible to provide an exhaustive list of that literature.39 Instead, we focus on a few papers and highlight their insights on the macroeconomic aspects of the housing market.40

One of the results of Kaplan et al. (2017) is that belief shocks matter considerably. In fact, the importance of beliefs has been discussed in the real estate literature, perhaps with different names.41 The more recent literature embeds “beliefs” in a general equilibrium search model so that we can quantitatively study its impact. For instance, Piazzesi and Schneider (2009) show that in a simple house search model, a small number of “optimistic” buyers in the economy can drive up housing prices because in a search environment, prices are determined in a bilateral trade. Optimistic buyers, although a small fraction in the economy, can arise endogenously as a large fraction of “active traders,” and hence are able to exert a large impact on the market price. The result is clearly in sharp contrast with the results reported in Kaplan et al. (2017), where “belief shock” needs to impact virtually all agents to generate a significant increase in housing prices. While the setup of the two models are very different and hence not trivial to compare, it is crucial to note that in Piazzesi and Schneider (2009), the housing market is decentralized and subject to search frictions, while in Kaplan et al. (2017), the housing market is centralized and cleared in every period.

Burnside, Eichenbaum, and Rebelo (2016) also explore the possibility of a small number of “optimists” that would affect the whole market through “social dynamics,” meaning that the “beliefs” of some economic agents could change when they meet others with different beliefs in a stochastic manner. Thus, a model with heterogeneous expectation is needed. Guided by the psychology literature, Burnside et al. (2016) assume that when two agents meet randomly, the agent who has less uncertainty about the fundamental of the housing market, measured by the entropy, would not be convinced by the agent who has more. However, the probability that an agent with high uncertainty is a decreasing function of the difference of the ratio of the uncertainty of the two agents that, other things being equal, agents are more likely to be changed by people with relatively similar beliefs than by people with very different beliefs.

Given these assumptions, the authors show that although initially there is only a small fraction of “skeptical” and “optimistic” agents, and most of the population is, therefore, “vulnerable,” that is, they are the ones who are most likely to change their beliefs, it is possible that the number of optimistic agents will rise and then fall as long as this uncertainty is not resolved. In one of the cases they consider, “skeptical” agents are the least likely to change, followed by the “optimistic” and “vulnerable” agents, who are most likely to change. Here, when an optimistic agent meets a skeptical agent, it is possible for the optimistic agent to be “converted” into a skeptical agent. However, when an optimistic agent meets a vulnerable agent, the former will remain optimistic and the latter may be converted into an optimistic agent. In a world of random matching, and the initial amount of skeptical and optimistic agents being equally small, the chance of an optimistic agent “converting” a vulnerable agent is much higher than the chance of him/her being “converted” by a skeptical agent. Hence, the population of optimistic agents is expected to grow. The population of skeptical agents will grow as well because whether they meet an optimistic agent or a vulnerable agent, they could “convert” them into skeptical agents. However, as the beliefs of skeptical agents are so different from those of the vulnerable majority, the growth rate of skeptical agents is slower than that of optimistic agents within certain parameter values. At some point, however, the relative population of optimistic agents becomes so large that the chance for them of meeting skeptical agents is higher than that of meeting vulnerable agents. The net growth of optimistic agents then becomes negative.

The rise and subsequent fall of the proportion of optimistic agents also leads to a boom–bust cycle of housing prices, as they, unlike the other types of agents, believe that housing will deliver a higher value of utility in the future. Burnside et al. (2016) demonstrate the basic mechanism with a simple model and then show that the intuition carries to a model with 12 different types of agents, who may differ in terms of their beliefs or their housing market participation. They calibrate their models and match some facts of the housing market.

While the previously mentioned papers are based on a random search, Hedlund (2016a, 2016b) explores a model using a directed search.42 In particular, Hedlund assumes that all transactions are intermediated by brokers. Hence, buyers buy from brokers and sellers sell to brokers. Modifying the “search activities” enables the model to consider a more complicated form of mortgage contracts than many search-theoretic models of housing.43 In Hedlund’s model, the pace of amortization is flexible and hence borrowers can slow down the speed of repayment when needed. However, homeowners can extract home equity only by paying off the original mortgage contract first and then originating a new mortgage contract.44 There are two issues to consider here. First, the homeowners who need to extract home equity may not have enough liquid assets to pay off the original mortgage contract. Second, even if they are willing to borrow, banks may not be willing to lend. In the model, banks need to pay for both the mortgage origination cost (which functions as a “fixed cost”) as well as serving cost to maintain a mortgage cost (which functions as a “variable cost”). Thus, banks may not want to originate a new mortgage contract in certain situations.

Equipped with this setup and through careful calibration, Hedlund mimics the stylized fact that leverage, selling price, and time-on-the-market (TOM) are correlated.45 The intuition is straightforward. When a homeowner needs to smooth out his/her consumption, he/she can either sell his/her liquid assets, extract home equity, or simply sell his/her house. When the number of liquid assets decreases, he/she may be tempted to extract the home equity. However, the bank may not want to originate a new mortgage contract for such needy homeowners. In this case, the homeowner will be forced to sell his/her house. Thus, homeowners with few liquid assets, so-called distressed owners, may be forced to launch a “fire sale.” However, those who are highly leveraged may not even be able to do that because they do not have enough liquid assets to make up the difference between the selling price and the outstanding mortgage loan. Therefore, those distressed owners who are also highly leveraged will be forced to post high selling prices. Undoubtedly, other things being equal, high selling prices will lead to a longer TOM, which further increases the opportunity for foreclosure. Hedlund (2016a, 2016b) articulates this intuition with a cleverly designed directed search model of the housing market.

Thus far, we have seen two extreme forms of house searching, namely, either completely random (such as the case of Burnside et al., 2016; Piazzesi & Schneider, 2009) or completely directed (such as the case of Hedlund), in which no seller sells to an end user directly. While these models succeed with respect to many dimensions, the fluctuations in terms of trading volume seem to have been overlooked. For instance, in Figures 5 and 6 of Burnside et al. (2016), we notice that the transaction volume (which is also the turnover rate, as the total supply of housing stock is fixed in their model) can deviate up to 20% from the steady-state value. Some studies are relatively silent on the turnover rate. Figure 1 represents the real house price and the turnover rate in the United States.46 To facilitate comparison, we normalize both series to 100 at the beginning of the period. It is clear that turnover rate fluctuates significantly more than the house price. Even at the aggregate level, the turnover rate at its peak can be double its own trough.

Figure 1. U.S. house price and turnover rate.

There are unanswered questions in Leung and Tse (2017). If we can have multiple equilibria in the housing market, we would want to know how equilibrium is being selected. A natural way to do so is to introduce the formation and evolution of beliefs. Thus, a natural extension is to combine a search model with beliefs (such as Burnside et al., 2016; Piazzesi & Schneider, 2009) with an explicit modeling of flippers (such as Leung & Tse, 2017) and examine both the theoretical and empirical implications. There are many opportunities for future research in this regard.

## Urban Development and Housing Market

According to the United Nations (2016), “Most people can agree that cities are places where large numbers of people live and work; they are hubs of government, commerce and transportation. But how best to define the geographical limits of a city is a matter of some debate. So far, no standardized international criteria exist for determining the boundaries of a city and often multiple different boundary definitions are available for any given city.” In the economics literature, much has been discussed on city formation, the interaction of spatial agglomeration and economic development, and related topics.48 Here, we only highlight a few papers, which may encourage more research in this area.

Most macro-housing papers assume that there is only one housing market, and abstract away the spatial and urban considerations.49 Wang and Xie (2014) develop a two-sector growth model with explicit spatial considerations. They attempt to explain four stylized facts within a unifying framework. First, housing structure outgrows housing. Second, house prices grow much slower than land rents. Third, both house price and land rent gradients are downward sloping away from the city centers. Fourth, cities with flatter population gradients also have flatter housing quantity and price gradients. More specifically, Wang and Xie assume that the city is a line segment and agents can choose their preferred locations. Hence, housing prices can vary across locations (i.e., different “points” on the same line segment) in equilibrium. In their model, housing depends on the amount of land as well as the physical structure. While land does not depreciate, the physical structure does depreciate over time. Hence, agents need to invest in structure continuously and consequently to trade off the investment in structure (for “housing consumption”) versus investment in physical capital (for goods production). They further deviate from the commonly used specification in the macro-housing literature in that (a) they assume that there is a positive amount of minimum structure, and (b) housing is a luxury good in the sense that its income inelasticity differs from that of consumption goods. As agents can reallocate themselves freely across locations, the prices must adjust in a way that the utility at each location is equalized. Their model can mimic some cross-sectional and time-series stylized facts. In addition, they show that if instead (a) they assume homothetic preference (i.e., housing has the same income elasticity as consumption goods), or (b) there is no minimum physical structure, then the model would generate counter-factual predictions. This is consistent with the previous literature on “unbalanced growth,” that considering non-homothetic preference is very important in explaining the long-term trend (e.g., Kongsamut, Rebelo, & Xie, 2001). In other words, if we were to account for both cross-sectional and time-series facts of housing, we need to reconsider the standard macro-housing models.

Herkenhoff, Ohanian, and Prescott (2017), among others, take the “spatial macro-housing models” (SMHM) to another level. It is well known that land-use regulations limit housing supply and affect house prices.50 Leung and Teo (2011) built one of the first multi-regional, dynamic, general equilibrium models and show that their model is qualitatively consistent with several stylized facts. However, they do not provide a careful calibration of the regional-level data. Herkenhoff et al. (2017) aggregate the 48 contiguous states of the United States into eight regions and model them separately, thereby allowing parameters to differ across regions. They find that if they could deregulate California and New York back to their 1980 levels of regulation, the “gain” in consumption would increase by up to 5%, which is a very significant number.

While most macro-housing models focus on the U.S. case, there are important exceptions. For instance, there is much debate on whether (a) the housing market in China is expanding too quickly, (b) it could collapse soon, and (c) if it indeed collapses, it could exert negative externality on other markets and countries.51 Garriga, Hedlund, Tang, and Wang (2017) contribute to the debate by building a dynamic model with “rural” and “urban/city” areas. In their model, continuous productivity growth in the manufacturing sector plays an important role because it widens the productivity gap between the manufacturing and agricultural sectors. It enables city workers to afford more expensive housing over time, and also attracts rural workers, who can only work in the agricultural sector, into urban areas. Both effects lead to a continuous increase in the relative price of housing. The authors calibrate their model carefully and find that they can explain most changes in the housing market. Clearly, China is not the only developing country. Therefore, there is much room for the application and extension of the studies in other developing housing markets.

## Urban Policy and Human Capital

Recent research confirms that human capital is important to economic growth, and educational reform has the potential to significantly improve the economic growth rate (e.g., Hanushek, Ruhose, & Woessmann, 2017a, 2017b; Hanushek & Woessmann, 2015). Research has also confirmed that early stage human capital formation (for instance, before the age of 10, or even earlier) is crucial to later stage human capital formation (e.g., Cunha & Heckman, 2007, 2008; Cunha, Heckman, & Schennach, 2010; Heckman, 2008). In the United States, the approach to providing local public financing of primary and secondary schools naturally binds the local housing market and the financing of pre-college education. Hanushek and Yilmaz (2011, p. 583) summarize it this way: “In simplest terms, poverty, race and schooling are very highly correlated with location. . .. The reliance on the local tax for a large portion of school funding implies that the government grant system has an important effect on both locational decisions and educational outcomes . . . Education in the United States is provided by local school districts that operate with considerable autonomy. Funding is provided by a combination of local, state, and federal revenues with the level of spending and the performance of schools varying significantly across school districts. . ..” Thus, it is possible that some families who can only afford less desirable neighborhoods raise their offspring in those neighborhoods. In turn, the future development of these offspring may be constrained and adversely affected, and hence result in cross-sectional segregation and intergenerational immobility. Benabou (1993, 1996a, 1996b) and Durlauf (1996), among others, explore such related theoretical possibilities. Empirically, however, this seems to be controversial and many authors have contributed to this literature.52 Here, we only highlight a few contributions and hopefully encourage even more research on this literature. For instance, Hanushek and Yilmaz (2007, 2013) build a static general equilibrium model, which allows for endogenous community choice (according to Tiebout) and spatial locational choice (according to Alonso).53 They calibrate the model to match some stylized facts of the United States. Based on the calibrated parameters, they point out that the manner in which education is financed matters. Hanushek, Sarpça, and Yilmaz (2011) introduce private schools in that framework and find that private schools can indeed improve the welfare of all types of agents. Leung, Sarpça, and Yilmaz (2012) consider public housing versus housing vouchers in the Hanushek–Yilmaz model. They find that although there are different forms of market imperfections in the model, the conventional wisdom, namely, distributing cash is better than distributing public housing units, prevails. Hanushek and Yilmaz (2015) find that the zoning policy affects both education and the housing market. Bayer, McMillan, Murphy, and Timmins (2016), however, estimate a dynamic, partial equilibrium model. They assume that the per-period utility function is somehow additively separable. Their dynamic estimates suggest that some prior estimates that are based on static demand functions of non-market amenities may be significantly biased. Given all of these calibrations and estimations, one may still wonder whether, or under what conditions, an urban policy (such as public housing or affordable housing policy) or an education finance reform (such as school finance consolidation) would necessarily improve the welfare and promote the accumulation of human capital and economic growth.54 Clearly, much more work is needed in this area.

# Concluding Remarks

To most households, housing is the most important form of asset holding that they can manage on their own (a retirement fund is clearly important as well but is typically subject to different constraints before retirement). Recently, the importance of housing has been increasingly recognized in economics and finance. This article achieves two objectives. First, we re-examine the “stylized facts” of macro-housing. With respect to the business cycle frequency, we find that in many cases, the correlations between traditional MV and HMV have weakened following the GFC. However, the correlations between macro-finance variables and HMV have strengthened following the GFC. In the medium-cycle frequency, some correlations display the same change as in the business cycle frequency while others do not. Therefore, we urge researchers to pay more attention to the “stylized facts” in macro-housing research.

Our second objective is to review some of the literature. We highlight some contributions from important topics: mortgage and bankruptcies, search and beliefs, urban development and housing market, and urban policy and human capital. We find that substantial progress has been made, and at the same time, there remain open questions in these areas. Clearly, many important papers and research areas have been overlooked. We can only allow for such “recognition deficit” for the moment and hope for a more elaborate survey in the future.

# Acknowledgments

The authors wish to thank all (former and current) co-authors, colleagues, teachers, conference discussants, and referees (whose identities are unknown to us) for all the comments, especially (in alphabetical order) Nan-Kuang Chen, Bob Edelstein, Eric Hanushek, Yuichiro Kawaguchi, Fred Kwan, Guannan Luo, Steve Malpezzi, Tim Riddiough, Martin Schneider, Jim Shilling, Chung Yi Tse, Ko Wang, Ping Wang, and Kuzey Yilmaz for sharing their many insights over the years, and City University of Hong Kong for financial support.

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# Appendix

Forbes and Rigobon (2002) show that with a change in variance over time, the measured correlation may be biased. We therefore follow Stock and Watson (2002) to standardize the variance. Our procedure is given below.

Step one. We first use the Christiano–Fitzgerald filter to extract the business cycle components in the PCSS. By construction, the extracted time series is stationary and hence moments are well defined. Let ${xt}$ be such a time series. The correspondence variance of ${xt}$ is denoted by $σx2$. We then “standardize” the volatility of all variables. Now a new variable $zt=xtσx$ is defined. Hence, $var(zt)=var(xt/σx)=var(xt)/(σx2)=1$. We apply this data transformation to all variables and hence all variables have the same volatility. We can then compute all of the needed correlations.

Step two. We repeat the same procedure on the full sample. Note that even for the same variable $xt$, for instance GDP, the sampling period is now different and hence the variance of the extracted series may also be different. A potentially different transformed variable ${z′t}$ will be generated. We apply the procedure to all variables and then re-compute all of the needed correlations.

Step three. We repeat both step one and step two, only this time we extract the medium-cycle components instead of the business cycle components. We calculate and tabulate all of the needed correlations.

## Notes:

(1.) This article focuses on housing. Related literature covers studies on land markets, which is beyond the scope of this article. See Davis and Palumbo (2008), Krainer, Spiegel, and Yamori (2010), Leung and Chen (2006), Liu, Wang, and Zha (2013), Ogawa, Kitasaka, Yamaoka, and Iwata (1996), among others, and the references therein.

(2.) The list is growing constantly, and it is difficult to provide a “complete list” here. See Bostic and Ellen (2014), Davis and Van Nieuwerburgh (2014), Hendershott, Hendershott, and Shilling (2010), Ihlanfeldt and Mayock (2015), Jordà, Schularick, and Taylor (2016), Malpezzi (2017a, 2017b, 2017c), McMillen (2011), Piazzesi and Schneider (2016), Sanders and Van Order (2011), among others, and the references therein.

(3.) See Baxter (1996), Greenwood and Hercowitz (1991), Kiyotaki and Moore (1997), among others.

(4.) See Aoki, Proudman, and Vlieghe, G (2004), Chang (2000), Davis and Heathcote (2005), Iacoviello (2005), Iacoviello and Minetti (2008), Jin and Zeng (2004), Kan, Kwong, and Leung, 2004), Kwong and Leung (2000), Leung (1999, 2003, 2014), Leung and Feng (2005), Leung, Lau, and Leong (2002), Leung, Leong, and Chan (2002), Lin, Mai, and Wang (2004), Ortalo-Magné and Rady (2006), Tse and Leung (2002), among others, and the references therein.

(5.) For an introduction to the U.S. housing market, see Green and Malpezzi (2003), among others. For a comparison of the U.S. mortgage system with that in other countries, see Green (2008), Green and Wachter (2005), Cho (2007, 2009), among others.

(6.) The literature is too vast to be reviewed here. See Bardhan, Edelstein, and Kroll (2012), Bardhan, Edelstein, and Leung et al. (2004), Bardhan, Edelstein, and Tsang (2008), Ben-Shafar, Leung, and Ong (2008), Cesa‐Bianchi, Cespedes, and Rebucci (2015), Chang and Chen (2012), Chao and Yu (2015), Cheung, Chow, and Yiu (2017), Edelstein and Kim (2004), Ho, Ma, and Haurin (2008), Kawaguchi (2009), Kim (2012), Leung (2017), Leung and Chen (2017), Leung and Quigley (2007), Leung and Tang (2012), Lum (2012), Malpezzi and Sa-Aadu (1996), Mera and Renaud (2000), Seek, Sing and Yu (2016), Seko, Sumita, and Naoi (2012), Tirtiroglu (2012), Tiwari (2001), among others, and the references therein.

(7.) The literature is too vast to be reviewed here. See Gau and Wang (1991), Geltner (1991), Geltner, MacGregor, and Schwann (2003), among others.

(8.) See Baxter (1991, 1994), Canova (1998), among others, who show that “stylized facts” are not robust to different filtering methods and the first-difference filter exaggerates the high-frequency components. Burnside (1998) shows that “stylized facts” are indeed robust if one turns to the frequency domain. Cogley and Nason (1995), King and Rebelo (1993) show that the Hodrick–Prescott (HP) filter creates spurious serial correlations and distorts “stylized facts.” Baxter and King (1999) and Christiano and Fitzgerald (2003) show that the band-pass filter, which builds on some results in the frequency domain, is statistically “superior” to the commonly used HP filter. In the present context, the “full sample” includes the GFC, which could be a high-frequency event for some data series, and at the same time a lasting event for other series. As we do not know a priori in which series the GFC is short-lived, and in which it is a long-lasting event, it is natural to use the band-pass filter to ensure that the corresponding periodicities of the components that we extract are precise.

(9.) Clearly, the treatment here is heuristic. For further treatment of the subject, see Berge (2015), Camacho and Perez-Quiros (2002), among others, and the references therein.

(10.) For a general discussion on recovery and labor markets, see Christiano, Eichenbaum, and Trabandt (2015, 2016), among others. For a discussion on the slow recovery and the housing markets, see Garriga and Hedlund (2017), Hedlund (2016a, 2016b), Luo (2017), among others, and the references therein.

(11.) For instance, see Hubbard and O’Brien (2014). See also Doepke and Schneider (2006), among others.

(12.) The literature is too vast to be reviewed here. See Del Negro, Giannoni, and Schorfheide (2015), Gilchrist, Schoenle, Sim, and Zakrajsek (2017), Hall (2011), King and Watson (2012), among others.

(13.) It has been studied extensively. See Greenwood and Hercowitz (1991), Chang (2000), Chen and Liao (2018), among others, and the references therein.

(14.) The literature is too vast to be reviewed here. See Bostic, Gabriel, and Painter (2009), Calomiris, Longhofer, and Miles (2009), Carroll, Otsuka, and Slacalek (2011), Case, Quigley, and Shiller (2005), Gerardi, Herkenhoff, and Willen (2018), Kaplan, Mitman, and Violante (2016), Zhou and Carroll (2012), among others, and the references therein.

(15.) There are studies relating the housing market to international trade, both theoretically and empirically. See Bardhan et al. (2004), Corrigan (2017), Leung (2001), Leung, Shi, and Tang (2013), among others.

(16.) See Aizenman and Jinjarak (2009), Tomura (2010), among others, and the references therein for related analysis.

(17.) The literature is too vast to be reviewed here. See Gust, Herbst, Lopez-Salido, and Smith (2017), Hirose and Inoue (2016), Wu and Xia (2016), among others.

(18.) See Sims (2010, 2012), among others.

(19.) The literature is too vast to be reviewed here. See Chang, Chen, and Leung (2010, 2011, 2012, 2013, 2016), Chen and Wang (2007, 2008), Jin, Leung, and Zeng (2012), Malpezzi (2017a, 2017c), Yavuz (2014), among others, and the references therein.

(20.) See Frost et al. (2015), Ihrig, Meade, and Weinbach (2015), Selgin (2018), among others.

(21.) The literature is too vast to be reviewed here. See Bhattarai and Neely (2016), Chodorow-Reich (2014), Gertler and Karadi (2011), Gertler and Kiyotaki (2011), among others, and the references therein.

(22.) See Bernanke and Gertler (1995), Bernanke, Gertler, and Gilchrist (1999), among others. Jin et al. (2012) modify the framework of Bernanke et al. (1999) and study how the EFP and housing price can interact.

(23.) The literature is too vast to be reviewed here. See Chang et al. (2012, 2013, 2016), Chu (2010), Davis and Martin (2009), Kwan, Leung, and Dong (2015), Leung (2007), Piazzesi, Schneider, and Tuzel (2007), Quan and Titman (1999), among others, and the references therein.

(24.) See Drehmann, Borio, and Tsatsaronis (2012), among others. Borio (2014, p. 183) claims “the financial cycle has a much lower frequency than the traditional business cycle …”

(25.) Research efforts have been devoted to explaining the heterogeneous “stylized facts” across different frequencies. See Pancrazi (2015), who shows how a change in the persistence of the exogenous shocks can generate “heterogeneous great moderation,” among others.

(26.) Clearly, the treatment here is heuristic. The literature on “money neutrality” is too large to be discussed here. See King and Watson (1997), Vaona (2015), among others, and the references therein.

(27.) See Baxter (1996), Chang (2000), Greenwood and Hercowitz (1991), among others, and the references therein.

(28.) See Benes, Kumhof, and Laxton (2014), Del Negro et al. (2013, 2015), Funke, Leiva-Leon, and Tsang (2017), Guerrieri et al. (2015), among others, for related efforts.

(29.) The literature is too vast to be reviewed here. See Chang et al. (2011, 2012, 2013, 2016), Chen, Chen, and Chou (2010), Chen, Cheng, Chu (2015), Chen, Chu, Liu, and Wang (2006), Chen and Leung (2008), Deng, Girardin, Joyeux, and Shi (2017), Phillips, Shi, and Yu (2015a, 2015b), Yiu, Yu, and Jin (2013), among others, and the reference therein.

(30.) The literature is too vast to be reviewed here. See Athreya, Sanchez, Tam, and Young et al. (2015, 2018), Athreya, Tam, and Young (2012), Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007), Corbae and D’Erasmo (2017), Duncan and Nolan (2018), Gertler and Kiyotaki (2011), Gertler, Kiyotaki, and Prestipino (2015), Khan, Senga and Thomas (2017), Krueger, Mitman, and Perri (2016), among others, and the references therein.

(31.) There are many related studies on mortgage contract and/or homeownership. See Amior and Halket (2014), Anagnostopoulos, Atesagaoglu, and Carceles-Poveda (2013), Fisher and Gervais (2011), Ghent (2012), Kydland, Rupert, and Sustek (2016), Luo (2017), Ortalo-Magne and Prat (2014), Ríos-Rull and Sánchez-Marcos (2008), Sun and Tsang (2017), Yao (2018), among others, and the references therein.

(32.) Some previous studies model a mortgage contract as a staggered contract, following Taylor (1980), and others model a mortgage contract as a down payment constraint, following Kiyotaki and Moore (1997). There is related literature on optimal mortgage contract design, which is typically partial equilibrium and theoretical in nature. See Pikorski and Seru (2018), among others, for a review of that literature.

(33.) See Karsten, Krueger, and Mitman (2013), Lambertini, Nuguer, and Uysal (2017), among others, for a related analysis.

(34.) See also Kaplan and Violante (2014), among others, for a related analysis.

(35.) See Chatterjee and Eyigungor (2015), among others, for a related analysis.

(36.) See Foote and Willen (2017), among others, for a related discussion.

(37.) This result is well known. See Dixit and Pindyck (1994), Stokey (2009), among others.

(38.) As a matter of fact, the “stylized facts” related to the Great Recession are evolving over time, as more data become available and more research efforts are being devoted. See Agarwal et al. (2017), Foote, Gerardi, Goette, and Willen (2008), Foote, Loewenstein, and Willen (2016), Foote and Willen (2017), Gerardi, Herkenhoff, Ohanian, and Willen (2015a), Gerardi, Lehnert, Sherlund, and Willen (2008), Gerardi, Rosenblatt, Willen, and Yao (2015b), Mian and Sufi (2015, 2016), Palmer (2015), among others, and the references therein.

(39.) Wheaton (1990) may be the first paper that adopts a search-and-matching approach (SMA) in an equilibrium model to study the house price and vacancy. SMA is then adopted by many authors, including Albrecht, Anderson, Smith, and Vroman (2007), Bayer, Geissler, and Roberts (2011), Díaz and Jerez (2013), Halket and Pignatti Morano di Custoza (2015), Halket and Vasudev (2014), Head and Lloyd-Ellis (2012), Head, Lloyd-Ellis, and Sun (2014), Head, Sun, and Zhou (2018), Hort (2000), Huang, Leung, and Tse (2018), Krainer (2001), Ngai and Tenrevro (2014), among others. The mathematical foundation of the search-and-matching models has been studied by Duffie and Sun (2012), among others.

(40.) The related literature uses partial equilibrium search models to analyze the housing market. Again, the literature is too vast to be reviewed here. See Anglin and Gao (2011), Arnott (1989), Deng, Gabriel, Nishimura, and Zheng (2012), Leung, Leong, and Wong (2006), Leung and Zhang (2011), Lin and Vendall (2007), Yavas (1992), among others, and the references therein.

(41.) The literature is too vast to be reviewed here. See Case and Shiller (1988, 2003), among others, and the references therein.

(42.) It is beyond the scope of this article to discuss the expanding literature on directed searching. See Wright, Kircher, Julîen, and Guerrieri (2017), among others, for a review of the literature.

(44.) See Laufer (2018), among others, and the references therein for the effect of equity extraction on the subsequent default probability.

(45.) Prior research explains the correlation by using behavioral economics such as the preference of loss aversion (e.g., see Genesove and Mayer, 1997, 2001). Some recent research, however, has cast doubts on the importance of loss aversion (e.g., see Easley and Yang, 2015 and Li, Seiler, and Sun, 2017).

(46.) Our real housing price is defined as the nominal house price, which is the transaction-based house price index from OFHEO (http://www.fhfa.gov) divided by the CPI from the Federal Reserve Bank at St. Louis. Transaction is measured by the quarterly sales in single-family homes, apartment condos, and co-ops, normed by the stock of such units. The sales data are from the Real Estate Outlook by the National Association of Realtors, compiled by Moody’s Analytics. The housing stock is defined as the sum of owner-occupied units and vacant for-sale-only units. The data are from the Bureau of Census’s CPS/HVS Series H-111.

(47.) See Guren (2016), among others, for more discussion on this point.

(48.) It is beyond the scope of this article to review the literature. See Berliant (2010), Berliant and Mori (2017), Berliant, Peng, and Wang (2002), Berliant and Wang (2008), Berliant and Watanabe (2015), Bond, Riezman, and Wang (2016), Duranton (2007), Duranton and Puga (2004), Glaeser (2011), Kopecky and Suen (2010), Li (2017), Liao, Wang, Wang, and Yip (2017), Liao and Yip (2018), Lucas and Rossi-Hansberg (2002), Schiff (2015), Wang, Wang, Yip, and Liao (2017), among others, and the references therein.

(49.) There are a few exceptions. See Lin et al. (2004), Leung and Teo (2011), among others, and the references therein.

(50.) The literature is too vast to be reviewed here. See Glaeser, Gyourko, Saks (2005), Green, Malpezzi, and Mayo (2005), Saiz (2010), among others, and the references therein.

(51.) The literature is too vast to be reviewed here. See Wu, Gyourko, and Deng (2012, 2016), Deng et al. (2017), Hsu, Li, Tang, and Wu (2017), Huang, Leung, and Qu (2015), Leung, Chow, Yiu, and Tam (2011), Song and Xiong (2018), Yang and Chen (2014), Zhang, Cai, Liu, and Kutan (2018), Zheng and Saiz (2016), among others, and the references therein.

(52.) The literature is too vast to be reviewed here. See Bayer, Ferreira, and McMillan (2007), Brasington and Haurin (2009), Epple and Nechyba (2004), Nechyba (2006), Nguyen-Hoang and Yinger (2011), among others, and the references therein. More recently, Chetty and Hendren (2018a, 2018b) confirm with their large administrative dataset that the neighborhood that children grow up in does have an impact on subsequent economic outcomes.

(53.) In Hanushek and Yilmaz, neighborhoods are identical ex ante. Other authors study the case when neighborhoods are different ex ante (e.g., downtown versus suburban). See de Bartolome and Ross (2003), among others, and the references therein.

(54.) For more discussion, see Disney and Luo (2017), Favilukis, Mabille, and Van Nieuwerburgh (2018), among others, and the references therein.