Pitfalls in VAR based return decompositions: A clarification

Abstract

We analyze the pitfalls involved in VAR based return decompositions. First, we show that recent criticism of such decompositions is misplaced and builds on invalid VAR models and erroneous interpretations. Second, we derive the requirements needed for VAR decompositions to be valid. A crucial – but often neglected – requirement is that the asset price needs to be included as a state variable in the VAR. In equity return decompositions this requirement is equivalent to including the dividend–price ratio in the VAR. Finally, we clarify the intriguing issue of the role of the residual component in return decompositions. In a properly specified first-order VAR, it makes no difference whether cash flow news or discount rate news is backed out residually, and it makes no difference whether both news components are computed directly or one of them is backed out residually.

Introduction

What causes asset prices to fluctuate? Is it mostly news about future cash flows or news about future discount rates (risk premia)? Since the seminal studies by Campbell, 1991, Campbell and Ammer, 1993, it has become standard to use VAR based return variance decompositions to answer this, and related, questions.¹ The methodology developed by Campbell and his coauthors (which builds on Campbell and Shiller’s (1988a) log-linear return approximation) is ingenious and has substantially increased our understanding of the underlying drivers of asset market fluctuations. However, there are a number of limitations and pitfalls involved in such VAR based decompositions, and although Campbell and Ammer, 1993, Campbell et al., 1997 caution against uncritical interpretation of VAR based decompositions, subsequent empirical studies often have not recognized the limitations and pitfalls involved and have computed invalid decompositions.

In a recent – but already widely cited – comprehensive study, Chen and Zhao (2009) discuss some of these limitations. In particular, they show that the decompositions can be quite sensitive to the variables included in the VAR, and they argue that the results are highly dependent on the predictive variables capturing the time-varying nature of expected returns. Of crucial importance here seems to be the treatment of cash flow news as a residual to be backed out from an identity. Based on their analyses, Chen and Zhao reach the very negative conclusion that VAR based return decompositions yield results that are “unrobust”, “not reliable”, and “counterintuitive”.

In this paper we provide further discussion of the various limitations and pitfalls involved in VAR based return decompositions. In particular, we take issue with some of the claims and analyses made by Chen and Zhao. First, we argue that Chen and Zhao’s interpretation of their decomposition for bond returns is wrong. Using excess bond returns they find a substantial residual component which they denote ‘cash flow news’ in their decomposition, and they claim that this is an error because for Treasury bonds nominal cash flows are fixed; thus, cash flow news must be zero. However, we point out that since Chen and Zhao make the decomposition for excess bond returns, i.e. bond return in excess of a short term nominal interest rate, what they call ‘cash flow news’ is in fact ‘nominal interest rate news’, and this component should not necessarily be zero. In addition, Chen and Zhao neglect the fact that the maturity of a bond shrinks over time. This introduces an additional error in their ‘cash flow news’ component. Thus, in contrast to what Chen and Zhao claim, their bond return decomposition does not document counterintuitiveness or unreliability of the VAR based decomposition approach.

Second, we point out a crucial aspect of VAR based decompositions neglected by Chen and Zhao, namely that in order for the decomposition to be valid, the asset price needs to be included as a state variable in the VAR. In equity return decompositions – due to the non-stationarity of stock prices – this requirement is equivalent to including the dividend–price ratio which is stationary. This is sometimes forgotten in empirical studies. In parts of Chen and Zhao’s analysis the price does not appear as a state variable, thus rendering those parts of their analysis invalid. If one discards these invalid analyses, the sensitivity problem found by Chen and Zhao is significantly reduced.

Third, and related to the first two points, an important element in most applications of the decomposition for equities is the treatment of cash flow news as a residual; dividend growth does not appear as a separate state variable in the VAR. This has caused confusion in the literature about the possible overstatement of the importance of the cash flow news component. Insufficient return predictability by the state variables is often conjectured to lead to an upward biased cash flow news component due to this component’s residual determination. However, as pointed out recently by Campbell et al. (2010), if one abstracts from the approximation error in the underlying log-linear return approximation, in a first-order VAR that contains the dividend–price ratio as a state variable, it does not matter for the return decomposition whether the cash flow news component or the discount rate news component is obtained as a residual. What matters is the choice of predictor variables (in addition to the dividend–price ratio) to include in the VAR.

Campbell et al. are not completely clear in explaining this insight, so we attempt to provide a full and detailed clarification of this point. We show that in order for the decomposition to be independent of which news component is treated as a residual, the underlying VAR model has to include the dividend–price ratio and the additional state variables have to be common to the computation of either return news or dividend news. We call this VAR model ‘proper’, and we show that in such a model the only difference between backing out the cash flow news component or the discount rate news component is the approximation error. We also emphasize that a ‘proper’ VAR model is not the ‘true’ model generating expected returns and/or dividends. No matter how rich the VAR information set is specified, it will always only be a subset of the full market information set. Thus, the result that a ‘proper’ VAR decomposition is independent of which news component is treated as a residual, does not require the included state variables to capture all predictable components of expected returns and dividend growth.

From the above point it also follows that in a properly specified VAR, it makes no difference whether return news and dividend news are both computed directly or one of them is backed out as a residual (this holds for a first-order VAR model; in higher-order models econometric complications arise due to near-perfect multicollinearity induced by the log-linear approximation).

Throughout, we illustrate our main points in empirical analyses using US stock market data. In order to make these points as clear and clearcut as possible, we intentionally keep these analyses very simple with only a few state variables included in the VAR models. Thus, our empirical analyses should not be seen as detailed and comprehensive investigations into the underlying drivers of stock market volatility. Our paper is mostly a methodological paper, but with clear implications for empirical practice.

The main conclusion from our analysis is that while Chen and Zhao are correct in arguing that the VAR based return decomposition approach is somewhat sensitive to the state variables included in the VAR, their claims about the “unreliability” and “counterintuitiveness” of the approach are unjustified. Used in a proper way, the approach is a useful and informative method for analyzing the movements and pricing of asset returns.

We emphasize that some of the points we make are not new and can be found in the existing literature. In our view, however, the way these points have been explained in the literature has not been completely clear, and it is our impression that many empirical researchers (ourselves included) have not been fully aware of the – in some cases quite intriguing – pitfalls in VAR based return decompositions and, hence, we believe that a detailed and comprehensive clarification will be useful.

The rest of the paper is organized as follows. Section 2 describes the basic return decomposition and the associated VAR methodology. Then, in Section 3, we explain in detail the various pitfalls involved in this methodology. We discuss decompositions for both bond and stock returns, and we provide empirical analyses to illustrate our main points. Section 4 offers some concluding remarks. The working paper version of this paper (which is available upon request) contains further analyses, elaborations, and discussions of the points we make.