اوراق قرضه در مقابل Q بازار سهام: آزمونی برای پایداری در سرتاسر فرکانس ها و طول زمان

In this paper we revisit the evidence recently provided by Philippon (2009) about the relationship among bond market's Q, stock market's Q and aggregate investments for the US. Specifically, we analyze the stability of the relationship between aggregate investment and the two measures of Q across frequencies and over time. We find that the relationship between aggregate investment and stock market's Q, in contrast to that with bond market's Q, is both frequency-dependent and time-varying. Both the successfulness of bond market's Q and the poor performance of the usual Tobin's Q can be explained by taking into account stability across frequencies of the first and instability over time of the latter.

Empirical work testing Q investment models generally adopt the stock market value of corporate equity as a proxy of the shadow value of capital. Consistent estimation of the Q model requires that the observable stock market valuation of a firm provides an accurate (true or, at least, error-free) measure of the present value of its expected future profitability, i.e. firm's fundamental value. Persistent divergences of stock market prices from their fundamental values are not unlikely given the nature and objectives of the different types of traders operating in the stock market and the forward looking nature of share prices which provide information on a firm's expected future value. Indeed, fluctuations of stock market valuations are mostly influenced by chartists or speculative investors: when investors with a short-term horizon predominate over long-term investors existing market valuations may be subject to divergence from firm's fundamentals for a very long time. The empirical failure of Tobin's Q investment equation based on the market value of equity has stimulated a number of different approaches (managers' and analysts' Q) 1 and measures 2 in order to overcome the difficulty of obtaining results consistent with the underlying theory ( Tobin, 1969). The literature on aggregate investment has recently shifted attention away from the stock market in favor of the bond market as a consequence of the disappointing empirical results of stock market's Q and the ability of credit spreads to forecast investment and output growth (e.g. Philippon, 2009 and Shen, 2010). In particular, Philippon (2009), starting from a yield-theory of investment where Tobin's Q is approximated by a linear function of the spread of corporate bonds over government bonds, shows that a bond market's Q measure performs much better than the usual Q measure in standard investment equations with post-war aggregate US data. Specifically, once bond market's Q is included as an additional regressor to the usual measure of Q the standard measure is no more significant. 3 Similar conclusions are obtained by Shen (2010) which, instead of getting a credit market Q from bond prices as Philippon (2009) does, tests the Q theory using the credit spreads directly. Indeed, his main findings are that the Q model is a good approximation and that the credit market may suffer less from mis-pricing than the equity market so that it may be possible to infer over/under-priced equity market valuation from credit spreads. The aim of this paper is to analyze the striking results obtained by Philippon (2009) through the time-frequency “lens” of wavelet analysis. Wavelet techniques can be used to gain information and insight about the characteristic features of the data, for example detecting the dominant scales of variation in the data, and to disentangle the individual effects of different variables at different time horizons. These properties of wavelets make them an attractive and powerful tool for the analysis of markets exhibiting multiscale features like financial markets, and stock markets in particular.4 Financial market prices are the outcome of the actions of different groups of agents, such as intraday, daily, short term and long term traders, each operating over a different time frame which ranges from seconds to years. These market participants are heterogeneous with respect to trading rules, beliefs, expectations, risk profiles, informational sets, and so on, and the interaction of these different types of traders and investors can generate very complex patterns in observed security prices (Hommes, 2006).5 Wavelets multi-resolution decomposition analysis, given its ability to separate out different time scales of variation in the data, can provide an effective solution to the analysis of processes with multiscale features. The key insight behind wavelet analysis is to analyze data at different scales or resolution levels. Indeed, wavelets provide a unique decomposition of time series observations that enables researchers to decompose the data in ways that are potentially revealing of relationships that are at best problematical using standard methods and aggregated data. In particular, wavelets' good frequency and time localization properties6 accord well with our needs, as most of the signals of practical interest in economics and finance have high frequency components for very short periods and low frequency components for long durations. We exploit the benefits of wavelet analysis as a complementary approach to classical analysis based on standard regression methods using continuous and discrete wavelet transform tools. Hence, we first explore the usefulness for exploratory data analysis of the tools associated with the continuous wavelet transform (CWT), that is the wavelet power spectrum and wavelet coherency. The first offers a powerful tool for detecting the dominant scales of variation in the data, and the latter a measure of the local correlation of two series in time-frequency space. The comparison between the time-frequency representations of the two measures of Q shows that while bond market's Q displays a very high coherency with aggregate investment throughout the sample at almost any scale (the only exception being the very short term scales), stock market's Q is closely related to investment rate only at scales corresponding to the long-run. Then, we perform a scale-by-scale regression analysis of the Q-relationship by applying the discrete wavelet transform, since regression analysis over timescale decompositions can provide important insights into the properties of economic relationships (see Ramsey, 1999). The results confirm the striking differences between the two measures of Q across scales previously evidenced by “wavelet-based” exploratory data analysis. The bond market's Q drives out traditional Q at any scale level, both jointly and in isolation. Moreover, when we test for equality of regression coefficients at different scale levels we are not able to reject the null hypothesis of coefficient's equality for bond market's Q at scales corresponding to frequencies larger than 2 years, the opposite being true for Tobin's Q. Finally, we consider an important assumption underlying the classical linear regression model, i.e. stability of regression coefficients which is rarely considered, but unlikely to hold when a given relationship is estimated over a long time span. Our results confirm previous findings in Holmes (2010) and Gallegati and Ramsey (2013) which suggest that variations in the Q-relationship over time are responsible for the poor performance of Tobin's Q investment equation. In addition, after taking into account for structural breaks in the Q-relationship the results indicate that in the long run stock market's Q is not only significant for aggregate investment, but has also an explanatory power marginally better than that of bond market's Q. The paper is organized as follows: in Section 2 we perform exploratory “wavelet-based” analysis of Philippon's dataset using the tools associated with the CWT. In Section 3 we perform a scale-by-scale regression analysis of the Q-relationship, while in Section 4 we examine the hypothesis of stability over time of the estimated relationship. Section 5 concludes the paper.

In this paper we revisit the evidence recently provided by Philippon (2009) about the relationship of aggregate investments with bond market's and stock market's Q for the US by exploiting the benefits of using the time-frequency “lens” of wavelet analysis for better exploration, evaluation and interpretation of relationship patterns between variables. Using continuous and discrete wavelet transforms tools we are able to provide important insights into the poor performance of the usual QT measure on one side, and the successfulness of bond market's Q on the other side. Firstly, our findings provide support to Philippon's explanation for the bond market's Q performing better than the usual measure of Q, namely that the bond market might be less noisy than the equity market because equity market can be subject to severe mis-pricing while the bond market is not, or at least, not as much. Indeed, after taking into account the effects of structural breaks in the Q-relationship we find that the mis-pricing effect has a prominent role in explaining the poor performance of Tobin's Q. The results at the longest scale are consistent with the hypothesis that the information content of stock market valuations tends to be more effective for investments in the long-run when existing market valuations are likely to better reflect the firm's fundamental value. Such a finding can reconcile the different puzzles characterizing the empirical literature on Tobin's Q, that is i) Q non-evidence, i.e. the lack of significance of QT regressor, and ii) non-Q evidence, i.e. the relevance of stock market for investment. Moreover, the results of “wavelet-based” exploratory and scale-by-scale regression analyses, with Philippon's bond-Q positively and significantly related to aggregate investment at almost any scale, indicate that the striking empirical performance of bond market's Q stems from the aggregation of variables displaying different relationships with aggregate investment at different time frames. To conclude, the hypothesis of a different information content of stock and bond market prices for investments at different horizons does not simply offer a more comprehensive view of aggregate investment, but also provides important implications for a multiscale approach to asset pricing. A theoretically plausible representation of the information content of asset prices and spreads about expected future returns and expected default risk, respectively, needs to consider the multiscale complex features of financial markets as evidenced by complementary rather than alternative effect of the information content of stock market's and bond market's prices for aggregate investment.