Markov-Switching Three-Pass Regression Filter

ABSTRACT

We introduce a new approach for the estimation of high-dimensional factor models with regime-switching factor loadings by extending the linear three-pass regression filter to settings where parameters can vary according to Markov processes. The new method, denoted as Markov-switching three-pass regression filter (MS-3PRF), is suitable for datasets with large cross-sectional dimensions, since estimation and inference are straightforward, as opposed to existing regime-switching factor models where computational complexity limits applicability to few variables. In a Monte Carlo experiment, we study the finite sample properties of the MS-3PRF and find that it performs favorably compared with alternative modeling approaches whenever there is structural instability in factor loadings. For empirical applications, we consider forecasting economic activity and bilateral exchange rates, finding that the MS-3PRF approach is competitive in both cases. Supplementary materials for this article are available online.

Key words:

• Factor model
• Forecasting
• Markov-switching.

ACKNOWLEDGMENTS

The authors thank the editor Todd Clark, an anonymous associate editor, three anonymous referees, Dalibor Stevanovic, James MacKinnon, seminar participants at Queen’s University, Banco de España and the 2016 conference of the Canadian Econometric Study Group held at Western University for helpful comments on a previous version of this article. The authors also thank Agustín Díaz for excellent research assistance. The views expressed in this article are those of the authors. No responsibility for them should be attributed to the Banco de España, the Eurosystem, the OECD or its member countries.

Notes

1 Groen and Kapetanios (2016) showed that partial least squares (and Bayesian methods) perform better than principal components when forecasting based on a large dataset with a weak factor structure. As partial least squares is obtained as a special case of the 3PRF (see Kelly and Pruitt 2015 for details), our method can also be adopted to introduce Markov switching in partial least-square regressions.

2 For example, extracting one factor from the MS-3PRF approach using a panel of 130 macroeconomic and financial variables with gross domestic product (GDP) growth as a target proxy takes about 350 sec using a laptop with a 2.7 GHz processor and 16 GB RAM.

3 In a Bayesian context, Guérin and Leiva-Leon (2017) developed an algorithm to estimate a high-dimensional factor-augmented VAR model with regime-switching parameters in the factor loadings to study the interactions between monetary policy, the stock market, and the connectedness of industry-level stock returns. See also Von Ganske (2016), who introduced regime-switching parameters in partial least-square regressions from a Bayesian perspective so as to forecast industry stock returns. Using a Bayesian framework, Hamilton and Owyang (2012) developed a framework for modeling common Markov-switching components in panel datasets with large cross-sectional and time-series dimensions to estimate turning points in U.S. state-level employment data.

4 Kelly and Pruitt (2015) specified Equation (1(1) $$\begin{array}{c}\hfill y_t={\beta }_0\left(S_{yt}\right)+\beta \left(S_{yt}\right){\mathbf{f}}_{t-1}+{\eta }_t,t=1,\dots ,T,\end{array}$$(1) ) as y_{t + 1} = β₀ + β₁f_t + η_{t + 1}, we use y_t as dependent variable to have a comparable timing of the Markov chains with the other model equations. Also, the lag structure in Equation (1(1) $$\begin{array}{c}\hfill y_t={\beta }_0\left(S_{yt}\right)+\beta \left(S_{yt}\right){\mathbf{f}}_{t-1}+{\eta }_t,t=1,\dots ,T,\end{array}$$(1) ) corresponds to the estimation step to calculate forecasts for y; for ease of exposition, we present the case of one-step forecasts, but the framework is straightforward to extend to the case of multi-step forecasts as we explain below.

5 Note that since proxies are used to compute the factors f_t, we need as many proxy variables as factors to be extracted, that is, k_f.

6 The variances of the errors in Equations (1(1) $$\begin{array}{c}\hfill y_t={\beta }_0\left(S_{yt}\right)+\beta \left(S_{yt}\right){\mathbf{f}}_{t-1}+{\eta }_t,t=1,\dots ,T,\end{array}$$(1) ) to (3(3) $$\begin{array}{c}\hfill x_{it}={\varphi }_{0,i}\left(S_{x_{i}t}\right)+{\varphi }_{f,i}\left(S_{x_{i}t}\right){\mathbf{f}}_t+{\varphi }_{g,i}\left(S_{x_{i}t}\right){\mathbf{g}}_t+{\epsilon }_{it},i=1,\dots ,N,\end{array}$$(3) ) are also allowed to be time-varying, with the time variation in each variance driven by the same variable-specific Markov chain that also drives changes in the parameters of the conditional mean of the corresponding variable.

7 This assumption is always satisfied if there is no Markov switching in (2(2) $$\begin{array}{c}\hfill z_{jt}={\lambda }_{0,j}\left(S_{z_{j}t}\right)+{\lambda }_j\left(S_{z_{j}t}\right){\mathbf{f}}_t+{\omega }_{jt},j=1,\dots ,k_{\mathbf{f}},\end{array}$$(2) ). Otherwise, its validity is an empirical issue that can be assessed by comparing the estimated Markov chains in regressions of each x_it either on the (estimated) factors or on the proxy variables z_t.

8 As it will be shown in Section 3, Monte Carlo experiments suggest a better performance, in finite samples, when using ${\widehat{\varphi }}_{B,it}$.

9 In a recursive forecasting exercise over a period T + 1, …, T + H, the forecast origin T would be recursively updated, as well as all the estimates in steps 1 to 3.

10 Forecasts for the predicted probabilities h-period ahead for h ∈ {1, 2, …} are calculated recursively as P(S_{yT + h} = j|Ω_T) = ∑^M_{i = 1}p_{y, ij}P(S_{yT + h − 1} = j|Ω_T).

11 The use of direct estimation for multi-step forecasting in nonlinear models is rather common, see, for example, Stock and Watson (1999), as it leads to major computational gains. Yet, the resulting direct forecasts are in general suboptimal with respect to the iterated ones, as the type of nonlinearity linking y_t and t − h dated explanatory variables is in general different from that linking y_t and t − 1 dated explanatory variables. In our context, the error term η_t in (11) can be serially correlated if the true model for y_t is Equation (1(1) $$\begin{array}{c}\hfill y_t={\beta }_0\left(S_{yt}\right)+\beta \left(S_{yt}\right){\mathbf{f}}_{t-1}+{\eta }_t,t=1,\dots ,T,\end{array}$$(1) ) and the factors are serially correlated, and this can affect the properties of the ML estimates of the states. An alternative forecasting procedure requires the introduction of a parametric model for the factors (e.g., a VAR), used to produce iterated forecasts for the ${\widehat{f}}_{T+h-1}$. Forecasts for y_{T + h} can be then obtained as ${\widehat{y}}_{T+h|T}={\sum }_{j=1}^M(P(S_{yT+h}=j|{\Omega }_T){\widehat{\beta }}_0(S_{yT+h}=j)+P(S_{yT+h}=j|{\Omega }_T)\widehat{\beta }(S_{yT+h}=j){\widehat{\mathbf{f}}}_{T+h-1})$. We leave an evaluation of this alternative forecasting approach for future research.

12 Ching, Fung, and Ng (2002) proposed a multivariate Markov chain approach for modeling multiple categorical data sequences.

13 The three-letter country codes follow the convention from the International Olympic Committee except for Taiwan, labeled as TAI.

14 Data for the euro before January 1999 and Taiwan were obtained from the U.S. Federal Reserve G.5 table (monthly average of daily data).

15 See http://www.bis.org/publ/rpfx16fx.pdf.

16 The Diebold and Mariano (1995) test of equal out-of-sample predictive accuracy is reported to give a sense of statistical significance of the point forecasting results. However, this test is based on the population MSFE (not the actual MSFE), so this test tends to reject the null of equal MSFEs too often.

17 Admittedly, in the case of the euro, the forecasting performance of the MS-3PRF (first and third passes) and the MSS-3PRF (first and third passes) approaches deteriorates as the forecast horizon lengthens, suggesting that it is not always relevant to model regime shifts in the forecasting equation.

18 Data descriptions and details on data transformation are available online at https://research.stlouisfed.org/econ/mccracken/fred-databases/Appendix_Tables_Update.pdf. The slight modifications we made to the original dataset are reported in the appendix.

19 Our results are qualitatively robust to the use of a different number of factors in the predictive equation. Detailed results are available on request.

20 When estimating the number of factors using information criteria, it is common to find a large number of factors summarizing the comovements of U.S. macroeconomic variables (e.g., McCracken and Ng (2016) estimated eight factors in the FRED-MD monthly macroeconomic database). However, in the forecasting exercise, in line with the literature, we use the first factor in the predictive equation. This corresponds to a real economic activity factor that closely follows the U.S. business cycle dynamics (see Figure 2). Using the first two factors in the predictive equation led to little changes in the forecasting performance.

21 The Bry and Boschan (1971) algorithm is a nonparametric method to estimate cycles in time series. We implemented the quarterly version of the Bry and Boschan (1971) algorithm from Harding and Pagan (2002), using the GAUSS code available at http://www.ncer.edu.au/resources/data-and-code.php.

22 We also compare the regime probabilities obtained with GDP growth as a target proxy with those estimated with the MS-3PRF factor as a target proxy. The regime probabilities are similar in both cases, which provides empirical support for our modeling assumption that the Markov chains in Equations (3(3) $$\begin{array}{c}\hfill x_{it}={\varphi }_{0,i}\left(S_{x_{i}t}\right)+{\varphi }_{f,i}\left(S_{x_{i}t}\right){\mathbf{f}}_t+{\varphi }_{g,i}\left(S_{x_{i}t}\right){\mathbf{g}}_t+{\epsilon }_{it},i=1,\dots ,N,\end{array}$$(3) ) and (5(5) $$\begin{array}{c}\hfill x_{it}={\varphi }_{0,i}\left(S_{x_{i}t}\right)+{\varphi }_i\left(S_{x_{i}t}\right){\mathbf{z}}_t+{ϵ}_{it},t=1,\dots ,T,\end{array}$$(5) ) are the same.

23 This result still holds when doing PCA at a monthly frequency and then aggregating the factor at a quarterly frequency.

24 As a side note, the first pass of the 3PRF filter could possibly accommodate mixed-frequency data using the techniques outlined in Foroni, Guérin, and Marcellino (2018); whereas, in the third pass of the filter, unrestricted mixed data sampling (MIDAS) polynomials could be used as in Hepenstrick and Marcellino (2016), and regime-switching parameters in the mixed-frequency predictive equation could be modeled as in Guérin and Marcellino (2013).