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High-dimensional dynamic factor models have their origin in macroeconomics, more specifically in empirical research on business cycles. The central idea, going back to the work of Burns and Mitchell in the 1940s, is that the fluctuations of all the macro and sectoral variables in the economy are driven by a “reference cycle,” that is, a one-dimensional latent cause of variation. After a fairly long process of generalization and formalization, the literature settled at the beginning of the 2000s on a model in which (a) both n, the number of variables in the data set, and T, the number of observations for each variable, may be large; (b) all the variables in the data set depend dynamically on a fixed, independent of n, number of common shocks, plus variable-specific, usually called idiosyncratic, components. The structure of the model can be exemplified as follows: (*) x i t = α i u t + β i u t − 1 + ξ i t , i = 1 , … , n , t = 1 , … , T , where the observable variables x i t are driven by the white noise u t , which is common to all the variables, the common shock, and by the idiosyncratic component ξ i t . The common shock u t is orthogonal to the idiosyncratic components ξ i t , the idiosyncratic components are mutually orthogonal (or weakly correlated). Last, the variations of the common shock u t affect the variable x i t dynamically, that is, through the lag polynomial α i + β i L . Asymptotic results for high-dimensional factor models, consistency of estimators of the common shocks in particular, are obtained for both n and T tending to infinity. The time-domain approach to these factor models is based on the transformation of dynamic equations into static representations. For example, equation ( ∗ ) becomes x i t = α i F 1 t + β i F 2 t + ξ i t , F 1 t = u t , F 2 t = u t − 1 . Instead of the dynamic equation ( ∗ ) there is now a static equation, while instead of the white noise u t there are now two factors, also called static factors, which are dynamically linked: F 1 t = u t , F 2 t = F 1, t − 1 . This transformation into a static representation, whose general form is x i t = λ i 1 F 1 t + ⋯ + λ i r F r t + ξ i t , is extremely convenient for estimation and forecasting of high-dimensional dynamic factor models. In particular, the factors F j t and the loadings λ i j can be consistently estimated from the principal components of the observable variables x i t . Assumption allowing consistent estimation of the factors and loadings are discussed in detail. Moreover, it is argued that in general the vector of the factors is singular; that is, it is driven by a number of shocks smaller than its dimension. This fact has very important consequences. In particular, singularity implies that the fundamentalness problem, which is hard to solve in structural vector autoregressive (VAR) analysis of macroeconomic aggregates, disappears when the latter are studied as part of a high-dimensional dynamic factor model.

The Hou–Xue–Zhang q-factor model says that the expected return of an asset in excess of the risk-free rate is described by its sensitivities to the market factor, a size factor, an investment factor, and a return on equity (ROE) factor. Empirically, the q-factor model shows strong explanatory power and largely summarizes the cross-section of average stock returns. Most important, it fully subsumes the Fama–French 6-factor model in head-to-head spanning tests. The q-factor model is an empirical implementation of the investment-based capital asset pricing model (the Investment CAPM). The basic philosophy is to price risky assets from the perspective of their suppliers (firms), as opposed to their buyers (investors). Mathematically, the investment CAPM is a restatement of the net present value (NPV) rule in corporate finance. Intuitively, high investment relative to low expected profitability must imply low costs of capital, and low investment relative to high expected profitability must imply high costs of capital. In a multiperiod framework, if investment is high next period, the present value of cash flows from next period onward must be high. Consisting mostly of this next period present value, the benefits to investment this period must also be high. As such, high investment next period relative to current investment (high expected investment growth) must imply high costs of capital (to keep current investment low). As a disruptive innovation, the investment CAPM has broad-ranging implications for academic finance and asset management practice. First, the consumption CAPM, of which the classic Sharpe–Lintner CAPM is a special case, is conceptually incomplete. The crux is that it blindly focuses on the demand of risky assets, while abstracting from the supply altogether. Alas, anomalies are primarily relations between firm characteristics and expected returns. By focusing on the supply, the investment CAPM is the missing piece of equilibrium asset pricing. Second, the investment CAPM retains efficient markets, with cross-sectionally varying expected returns, depending on firms’ investment, profitability, and expected growth. As such, capital markets follow standard economic principles, in sharp contrast to the teachings of behavioral finance. Finally, the investment CAPM validates Graham and Dodd’s security analysis on equilibrium grounds, within efficient markets.