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Empirical identification of factor models

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Abstract

In the conventional factor-augmented vector autoregression (FAVAR), the extracted factors cannot be used in structural analysis because the factors do not retain a clear economic interpretation. This paper proposes a new method to identify macroeconomic factors, which is associated with better economic interpretations. Using an empirical-based search algorithm, we select variables that are individually caused by a single factor. These variables are then used to impose restrictions on the factor loading matrix, and we obtain an economic interpretation for each factor. We apply our method to time-series data in the USA and further conduct a monetary policy analysis. Our method yields stronger responses of price variables and muted responses of output variables than what the literature has found.

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Notes

  1. For example, see Mumtaz and Surico (2009), Boivin and Giannoni (2010), Baumeister and Liu (2013), Dave et al. (2013).

  2. For example, in the term structure literature it is common to relate latent factors with “level,” “slope,” and “curvature” of the yield curve (Diebold et al. 2006; Dewachter and Marco 2006).

  3. For other related works that apply graph-theoretic causal search algorithm in the VAR, see Bessler and Lee (2002), Demiralp and Hoover (2003), Demiralp et al. (2008, 2009, 2014), Hoover (2005), Moneta (2008), Moneta and Spirtes (2006), and Phiromswad (2014).

  4. For example, see Bai and Ng (2007), Forni et al. (2000, 2009), Forni and Gambetti (2010), Giannone et al. (2005, 2006), and Hallin and Liska (2007).

  5. One important exception is Francesco and Milani (2006) who identify factors by categorizing observed variables based on conventional wisdom (e.g., real activity, inflation, and financial market) and extracting one latent factor from each category to be further used in monetary FAVAR analysis. For other related work that aims to identify different aspects of factors, see Boivin et al. (2009), and Reis and Watson (2010).

  6. The definition of factor loading here follows the notation of Bai and Ng (2013), which contains the time dimension through redefining the factors and observed data in matrix form instead of vector. For more details, see the original paper.

  7. Detailed discussion of the graph-theoretic causal search methodologies can be found in Spirtes et al. (2000) and Pearl (2000).

  8. For simplicity, we assume that there is no dynamic structure as in Eqs. (3) and (4).

  9. In order to apply the identification scheme proposed by Bai and Ng (2013), we need to know which variables can be considered as “pure variables.” If variables are reordered as \(X_{1}\), \( X_{3}\), \(X_{4}\), \(X_{2}\), and \(X_{5}\), the factor loadings matrix would take the form consistent with the restrictions in Eq. (4). What is needed is the knowledge that \(X_{1}\), \(X_{3}, X_{4}\) and \(X_5\) are pure variables. This is the purpose of our proposed search algorithm.

  10. Originally, Pearl (1988) studied only recursive structural equations. Spirtes (1995) and Koster (1996) independently illustrated that Pearl’s d-separation theorem can also be extended to nonrecursive structural equations.

  11. We assume the faithfulness assumption (Spirtes et al. 2000, p. 31) throughout the paper, which is a common assumption in the graph-theoretic causal search methodologies. A causal graph satisfies the faithfulness assumption if all conditional independence conditions entailed in the data-generating process are consistent with those obtained from Pearl’s d-separation theorem. However, it is important to note that there are examples where the faithfulness assumption is violated (see Sprites et al. 2000, p. 41; Pearl (2009), p. 62–63; Hoover 2001, p. 45–49, 151–153, 168–169).

  12. We assume that the data-generating process is linear which makes the tests of conditional correlation equivalent to the test of conditional independence.

  13. If all variables are correlated over time but not through the same factors, and we do not apply the filtering, none of the pure variables can be identified since no edges will be removed in step 2b.

  14. Stock and Watson (2002) prove the consistency of the estimated parameters in a factor-augmented regression. We test conditional correlations by partitioned regression (a residual-based approach), which is equivalent to testing conditional correlations of the observed variables when lagged factors (estimated from PCA) are always included in the conditioning sets. Therefore, the result of Stock and Watson (2002) applies in our context as well.

  15. We note that the number of factors used to filter the variables does not have to exactly match the number of factors employed in identifying the factors. This is because the purpose the filtering process is to remove dynamic dependencies among variables rather than performing structural analysis. We will return to this point in the later empirical analysis.

  16. The abbreviation “PC” stands for Peter and Clark, the first names of Spirtes and Glymour who invented this search algorithm.

  17. See Demiralp and Hoover (2003) footnote 8 for more discussion.

  18. This can also be illustrated based on the d-separation theorem of Pearl (1988). Two variables are correlated (or d-connected) since there is a directed path from the two variables through a common cause (which is the unobserved factor in this case).

  19. In principle, there are many ways we can do step 2b. Treating Eqs. (1)–(4) as the exact data-generating process, we can conduct step 2b based on testing just the unconditional correlations among the filtered variables. However, we choose to adopt the PC algorithm of Spirtes et al. (2000), which consider testing conditional correlations as well. We believe this is more appropriate because i) it would encompass testing unconditional correlations alone and ii) it allows for more parsimonious causal structure. For example, if the observed variables are also generated by other observed variables (i.e., not just the unobserved factors), then testing conditional correlations would be appropriate to eliminate an association among two variables. The PC algorithm is considered to be the most widely used graph-theoretic causal search algorithm (Demiralp and Hoover 2003).

  20. For example, a pure variable that conceptually belong to the output category but is not correlated with the total industrial production index would not be a credible candidate for the output factor.

  21. The only change from BBE’s categorization is the interest rate category. We distinguish levels and spread because these may capture different macroeconomic factors.

  22. We replace observations with absolute median deviations larger than six times the interquartile range with the median value of the preceding five observations.

  23. This number is the same as the one in Bernanke et al. (2005). We experimented with different lag lengths, but the result is almost unaffected.

  24. This is the suggested value for the PC algorithm of Spirtes et al. (2000) based on simulation experiments with similar sample size. We also tried with other values of \(\alpha \) (0.1, 0.025, and 0.01), but they did not affect the result in a significant way.

  25. This provides an informal support to our identification strategy in general, since the more the edges are concentrated within a group of variables, the higher the chance to detect a pure variable that is exclusively caused by a single factor.

  26. For example, within the price category three of the nineteen price indexes are not significantly correlated with the pure variable (PPI: Finished goods). However, this result was partially caused by how we grouped our price variables in the first place. If we treat PPI and CPI as two different categories, then PPI: Finished goods would be chosen as the pure variable in the first category, whereas CPI: All items less medical care would be chosen as the pure variable in the second category. Such distinction would increase the relevance of pure variable within the narrower subcategory.

  27. For example, Hallin and Liska (2007) suggest between one and four, depending on the sample period. Giannone et al. (2005, 2006) use two factors, whereas Bai and Ng (2007) and Forni and Gambetti (2010) use four factors in their benchmark analysis.

  28. We have also experimented with K = 3 for BBE (not shown), which coincides with what the authors call “preferred specification”. Raising the number of principal component makes the price fall slightly faster, but it has no effect on the responses of output and interest rate variables.

  29. According to NBER, the average peak-to-trough duration during recessions that occurred after 1960 was 11.6 months with the longest contraction observed from December 2007 to June 2009 (18 months).

  30. See Clarida et al. (1999); Estrella and Fuhrer 2003); Hoover and Jordá 2001); Demiralp et al. 2014) for more discussion.

  31. For the 124 variables case, the total time to complete the search process is close to 100 h, whereas with 50 variables the total time reduces to 12 h.

  32. Ahmadi and Uhlig (2009) and Forni et al. (2009) have used their own proposed methods to identify the structural shocks from the large dataset.

  33. Based on an example given in Spirtes et al. (2000), engine’s status (E) is a collider between the state of a car’s battery (B, dead or not) and the state of a car’s tank (T, with or no gas). Knowing that a car did not start, dead battery will tend to occur when the tank has some gas (and vice versa). Thus, the two variables (B and T) become dependent conditioning on the common effect or E (i.e., the collider).

  34. Downloaded on August \(28^\mathrm{th}\), 2012 from http://research.stlouisfed.org/fred2/.

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Correspondence to Piyachart Phiromswad.

Appendices

Appendix A: Graph-theoretic causal search methodologies

The objective of graph-theoretic causal search methodologies is to learn about causal structures from information regarding probabilistic distribution of the data-generating process estimated from the data. This paper focuses mainly on a system of linear structural equations with independent errors. We also make a distinction that some variables in a system are observed while some are latent variables. A causal graph can be used to represent such a system. X is a direct cause of another variable Y (represented as \(X\rightarrow Y)\) in a causal graph G if and only if there is a nonzero coefficient associated with variable Y in the equation that X is the dependent variable (for example, an equation \(X=\alpha Y+e\) with \(\alpha \ne 0\) and e as a structural error term). Thus, a causal graph illustrates the underlying zero and nonzero restrictions in a system of linear structural equations. We also say that, in the above case, X is a parent while Y is a child.

A directed path from X to Y exists if and only if there is a series of directed edges (or a directed edge) pointing in the same direction from X to Y. In this case, we also say that X is an ancestor while Y is a descendant. By convention, every variable is its own ancestor. On the other hand, there is an undirected path from X to Y if and only if there is a series of directed edges (or a directed edge) from X to Y regardless of the direction implied by the arrow. A vertex is a “collider” in an undirected path if and only if there are two directed edges pointing into this vertex in the path (i.e., in a graph \(X\rightarrow Y\leftarrow Z\), Y is a collider).Footnote 33 There are two types of collider; shielded and unshielded collider. For a shielded collider, the two directed edges are shielded by an edge. For an unshielded collider, the two directed edges are not shielded.

Pearl (1988)’s d-separation theorem states the following

Pearl (1988)’s d-separation theorem (as presented in Spirtes et al. 2000 p. 44): Let V be a set of all variables of a causal graph G in which X, Y, and Z be distinct subsets of variables in V. X and Y are d-separated given Z (i.e., independent conditional on Z) if and only if, there is no undirected path U between X and Y such that (i) every collider on U has a descendent in Z, and (ii) no other variable on U is in Z; otherwise X and Y are d-connected given Z (i.e., dependent conditional on Z).

For example, in a causal graph \(X\rightarrow Y\rightarrow Z\), X and Z are dependent unconditionally but become independent conditional on Y since Y is other variable which is on the only undirected path from X to Y.

Appendix B: Proof of proposition

Based on the d-separation theorem of Pearl, we provide the proof of the proposition in the text.

Restatement of Proposition

Let \(X_1 \) and \(X_2 \) be two distinct variables that are generated by Eqs. (1)–(4). If two filtered variables \( \tilde{X}_1 \) and \(\tilde{X}_2 \) are uncorrelated, then \(X_1 \) and \(X_2 \) do not share the same unobserved factor contemporaneously.

Proof of Proposition

Part 1: If two filtered variables \( \tilde{X}_1 \) and \(\tilde{X}_2 \) are uncorrelated, then \(X_{1}\) and \(X_{2}\) are also uncorrelated contemporaneously. To verify this by contradiction, suppose that \(X_{1}\) and \(X_{2}\) are also correlated contemporaneously. Then, there must exist an undirected path which make \(X_{1}\) and \(X_{2}\) d-connected. Since \(\tilde{X}_1 \) and \(\tilde{X}_2 \) are also influenced by the same factor loadings and the \(A_{0}\) matrix which encodes the contemporaneous causal relationship among the unobserved factors (\(F_{t})\), then \( \tilde{X}_1 \) and \(\tilde{X}_2 \) must be correlated which is a contradiction.

Part 2: If \(X_{1}\) and \(X_{2 }\)are uncorrelated contemporaneously, then \(X_{1}\) and \(X_{2}\) do not share the same unobserved factor contemporaneously. To verify this by contradiction, suppose that\( X_{1}\) and \(X_{2 }\)share the same unobserved factor contemporaneously (i.e., sharing a common cause). By direct application of Pearl’s d-separation theorem,\( X_{1}\) and \(X_{2}\) must be d-connected which is a contradiction.

Thus, if \(\tilde{X}_1 \) and \(\tilde{X}_2 \) are uncorrelated, then \(X_{1}\) and \(X_{2}\) do not share the same unobserved factor contemporaneously. Q.E.D.

Appendix C: Data

For the 124 variables case, series were taken from the Federal Reserve Economic Data (FRED) database, Federal Reserve Board (FRB), Bureau of Economic Analysis (BEA), US Census (Census), Global Financial Database (GFD). The data source for individual series is given in the last column of the data description. Prior to applying search algorithm, stationary transformations were applied based on the Augmented Dicky–Fuller unit root test with the optimal lag length chosen based on the Schwarz BIC. The type of transformation applied for individual series is given in the third column of the data description. The number in the column represents the following transformation: 1—no transformation; 2—first difference; 4—logarithm; 5—first difference of logarithm; 6—second difference of logarithm.

For the 50 variables case, all series were taken from the Federal Reserve Economic Data (FRED) database.Footnote 34 These variables have an asterisk next to the mnemonic in the data description. The number of variables is balanced among the following five broad categories so that the panel covers the major subset of the Federal Reserve’s information set: output (8 variables), labor market (12 variables), expenditure (8 variables), nominal variables (11 variables), and financial market variables (11 variables). Within each concept, the individual variables were chosen based on the “popularity” of variable among the FRED users.

Data description Description of data: 124 variables

Appendix D: FAVAR estimation

To identify factors using our method, we use three pure variables which are (a) IP: Durable consumer goods, (b) PPI: Finished goods, and (c) Effective federal funds rate. The first two variables are selected based on our search algorithm, whereas the effective federal funds rate is chosen so that it represents the monetary policy measure that are widely accepted in the literature. When running the VAR with the factors, we recursively order them as (a)\(\rightarrow \)(b)\(\rightarrow \)(c) within the same period. Finally, the monetary policy shock in the first period is set so that it is equivalent to a 25 basis point increase in the policy rate.

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Phiromswad, P., Yagihashi, T. Empirical identification of factor models. Empir Econ 51, 621–658 (2016). https://doi.org/10.1007/s00181-015-1025-9

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