U.S. stock markets and the role of real interest rates

The Quarterly Review of Economics and Finance 59 (2016) 231–242

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U.S. stock markets and the role of real interest rates Wanling Huang ∗ , André Varella Mollick, Khoa Huu Nguyen Department of Economics and Finance, The University of Texas – Pan American, 1201 West University Drive, Edinburg, TX 78539-2999, USA

a r t i c l e

i n f o

Article history: Received 25 October 2014 Received in revised form 13 July 2015 Accepted 26 July 2015 Available online 1 August 2015 JEL classification: F31 G15

a b s t r a c t Using weekly data from January 3, 2003 to March 27, 2015, we examine the responses of U.S. stock returns (S&P 500, DJIA, and NASDAQ) to monetary policy, controlling for WTI oil prices and the value of the U.S. dollar (USD) against major currencies. Based on differences between the federal funds rate and inflation expectations, U.S. real interest rates have become continuously negative since January 28, 2009. Vector auto-regressions (VARs) suggest stronger linkages more recently and vine copula models identify the structure of dependence across these markets, which can help investors optimize portfolio diversification. © 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.

Keywords: Exchange rates Oil prices Real interest rates U.S. stock markets Vine copulas

1. Introduction The severity of the most recent U.S. recession of 2008–2009 has led to a combination of expansionary fiscal and monetary policies by government and central bank alike. On monetary policy, in particular, in addition to the very low federal funds rate the U.S. Federal Reserve has put forward a monthly USD 85 billion bond-buying program aimed at keeping long-term interest rates low.1 Following the theoretical model by Chen, Roll, and Ross (1986), current stock prices reflect expected cash flows (earnings) discounted by the appropriate interest rates. Very low interest rates make the discounted cash flows high, thus justifying the increases in current stock prices. U.S. stock markets have indeed been moving upwards (with very high rates of return) since bottoming out in March 2009. The U.S. FOMC September 2013 decision illustrates well the typical market reaction right after the announcement: “On Wednesday, the

∗ Corresponding author. Tel.: +1 956 665 2825; fax: +1 956 665 5020. E-mail address: [email protected] (W. Huang). 1 One year into the recession, in early December of 2008, the effective Federal Funds rate was moved down to 0.12% on December 5, following levels of 0.52% on December 1 and 1.04% on October 15. From December 2008 onwards, the rate remained at the current very low levels within the 0.06–0.25% range. On October 15, 2013 it stayed at 0.10% using daily data from the U.S. Federal Reserve of St. Louis at http://www.research.stlouisfed.org/fred2/categories/118. In addition, Quantitative Easing programs were established in steps: QE1, QE2, and QE3.

Federal Reserve gave the markets uncertainty and confusion about plans to wind down its bond-buying program, and markets loved it, sending U.S. stock indexes to records. The Dow Jones Industrial Average rose 147.21 points, or 0.9%, to 15,676.94, a closing high. Bond prices notched their strongest gain since November 2011. Commodity prices jumped, and foreign stocks benefited even more than U.S. shares. The celebration was for the short term, based on the Fed’s decision to surprise investors with the news it wouldn’t begin reducing its bond-buying program after all.” (The WSJ, September 19, 2013). For the whole of 2013, stocks have returned 27% (DJIA) and 30% (S&P 500), making the year one of the best ever for equity markets. This paper estimates U.S. stock market responses to monetary policy, allowing for oil prices and exchange rates conveying important information from other financial markets around the world. While recent research by Laopodis (2013) suggests varying U.S. stock market responses to the FED conduct of monetary policy, we allow in this paper for a combination of nominal interest rates and price pressures in goods markets as the driving forces, controlling for other financial markets. We pay particular attention to the extremely low levels of nominal interest rates in the U.S., which has made the real interest rate become negative. We identify two subsamples to test changes in the relationships among the series over time by picking up the date in which real interest rates have become negative (on January 29, 2009). According to Fig. 1, there is a period before the 2008–2009 when real interest rates dropped below zero. That was short-lived, however, recovering afterwards

http://dx.doi.org/10.1016/j.qref.2015.07.006 1062-9769/© 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.

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W. Huang et al. / The Quarterly Review of Economics and Finance 59 (2016) 231–242

Fig. 1. Movements of the expected inflation and ex-ante real interest rates.

to the more normal positive rate, representing a greater than zero marginal product of capital. Following the intensity of the crisis, the real interest rate became negative and remains negative at the time of writing. Having the marginal product of capital become negative in the U.S. for more than six years has probably helped equity markets in the short-run. In addition, it may also have changed dramatically the nature of how stock prices respond to major financial markets. It is important, however, to allow for cross-market financial linkages. The empirical literature on commodity and financial markets contains a variety of established results. Vector autoregressions (VAR) by Cologni and Manera (2008) suggest that policymakers adjust interest rates in response to oil price shocks. Theories based on the opportunity cost of oil extraction and storage suggest that falls in real interest rate results in lower production (and higher prices) and vice versa.2 Akram (2009) finds that oil prices increase with negative movements in U.S. real interest rates. In his quarterly VAR model from 1990:1 to 2007:4 with OECD industrial production, the real interest rate, effective real exchange rate, and real price of oil, shocks to the real interest rate account for more than 20% of the forecast error variance in oil prices and real exchange rate fluctuations account for a little lower than 20%. Arora and Tanner (2013) revisit this conjecture for monthly frequency VAR from 1975:1 to 2012:5 and conclude that oil prices have become more responsive to long-term U.S. and international real interest rate after 2000. As for the link between oil and exchange rates in the long-run, Lizardo and Mollick (2010) find that oil prices significantly explain movements in the value of the U.S. dollar (USD) against major currencies from the mid-1970s to 2007:12 (with varying start dates), while Beckman and Czudaj (2013) find cointegration in monthly data from Jan 1974 to Nov 2011 between oil WTI prices and the real broad index (USD versus 26 currencies), as well as with the major index (USD versus 7 major currencies). Since an expansion of domestic money depreciates the USD against other currencies, lower U.S. interest rates suggest (by the

2 Akram (2009) discusses the no-arbitrage condition in detail. There are at least three channels in which the real interest rates inversely affect commodity prices. First, low real interest rates increase the price of storable commodities by increasing the incentive for extraction tomorrow rather than today. When the today supply reduces, the price of the commodities will increase. Second, low real interest rates allow firms to store more commodities. When the demand for a commodity is increasing, its price will rise. Third, when the real interest rate is too low, investors tend to shift their investment from T-bill into commodities, causing increases in the demand for commodities, thus the price.

UIP condition) that the rate of depreciation of the USD must fall. It is thus important to control for exchange rate effects when verifying the links between real interest rates and oil prices and between real interest rates and stock prices. Of particular interest is the period when the FED started expanding its balance sheet to handle the financial crisis. There are, of course, many ways to identify changes in monetary policy. We will focus in this paper on a market-driven indicator based on when the U.S. real interest rate became negative for a substantial amount of time, which has been recently used by Huang, Mollick, and Nguyen (2015) on a study of disaggregated commodities (copper, cotton, gold and oil) and the value of the USD; yet without considering stock markets. There are three main reasons for revisiting the attention to equity markets for the particular sample used in this paper. First, there is a vast literature developed by Campbell and Ammer (1993) and Thorbecke (1997) for U.S. data and extended to international stock markets by Campbell (1998), which focus on a time period of positive real interest rates for postwar economic data. Second, for the cyclical behavior of the Kydland–Prescott economy reported in Prescott (1986), the standard deviation of U.S. GNP is 1.79%, of hours is 1.23% and of real interest rate is only 0.22%, which makes it considerably less volatile than output. Since the real interest rate can be interpreted as the marginal product of capital, it is interesting to know if the present value discounted model of stock prices behaves differently under positive and negative (real) returns. Third, examining relatively symmetric subperiods of volatile versus always negative real interest rates in this sample (317 and 322 weekly observations, respectively) allows a finer comparison of monetary policy forces against foreign exchange and commodity prices. Using weekly data from January 3, 2003 to March 27, 2015, we examine the responses of U.S. stock returns (S&P 500, Dow Jones, NASDAQ) to monetary policy, controlling for WTI oil price returns and the value of the U.S. dollar (USD) against major currencies. Correlation coefficients show very different co-movements between the two periods and VARs suggest stronger linkages in the more recent period. In particular, we find for stock markets that they respond – in the more recent period – negatively to both real interest rates (and to the value of the dollar) and positively to increases in oil price shocks. Also, the magnitudes of the responses are larger in the period of financial crisis and Quantitative Easing. Vine copula models complement the above time series methods by looking at the dependence pattern among the four financial markets. One of the most remarkable features of copulas is that they capture the whole dependence structure between variables and not only the linear correlation between them. The vine copula methodology was proposed by Aas, Czado, Frigessi, and Bakken (2009), based on Joe (1996) and Bedford and Cooke (2001, 2002), and developed further by many articles in the recent literature. See, e.g., Dißmann, Brechmann, Czado, and Kurowicka (2013) and references therein, Min and Czado (2014), Beare and Seo (2015), Brechmann and Joe (2015), Schepsmeier (2015), Weiß and Scheffer (2015), etc. Vine copula has been widely applied in finance, see, e.g., Low, Alcock, Faff, and Brailsford (2013), Weiß and Supper (2013), Abbara (2014), Arreola Hernandez (2014), Brechmann, Czado, and Paterlini (2014), Markwat (2014), Allen, McAleer, and Singh (2014), Zhang (2014), Siburg, Stoimenov, and Weiß (2015), etc. For example, it can be applied in portfolio optimization in mainly two directions. Efficient diversification of investments based on the mean-variance analysis of Markowitz (1952) is widely used, however, its normality assumption does not usually fit the data in finance. Mendes, Mendes Semeraro, and Cmara Leal (2010) thus propose a robust vine copula mean-variance method, which is applied in Mendes and Marques (2012) and Arreola Hernandez (2014). In order to catch skewness and asymmetric dependence for asset allocation, Patton (2004) proposes a new method of portfolio optimization in the bivariate case using copulas, which can be easily extended to the multivariate

W. Huang et al. / The Quarterly Review of Economics and Finance 59 (2016) 231–242

case using vine copulas. Riccetti (2013) and Wei et al. (2013) tried application in this direction, which of course can be explored much more. In this paper, we use vine copula models to identify the structure of dependence across the four financial markets which can at least help investors optimize portfolio diversification. Vine copula has also been applied in exchange rate returns, see, e.g., Czado, Schepsmeier, and Min (2012) and Loaiza Maya, Gomez-Gonzalez, and Melo Velandia (2015). However, to our knowledge, this is the first study that analyzes the real interest rate – stock market link by using vine copula models. The data and methodologies are described in Section 2, empirical results are displayed in Section 3, and Section 4 concludes the paper. 2. The data and methodologies We investigate the relationships among the financial markets by looking at weekly data on oil price returns, Trade Weighted U.S. Dollar Major Currencies Index returns, changes in implied real interest rate and major U.S. market indices’ returns.3 Our weekly data ranges from January 3, 2003 (Friday) to March 27, 2015 (Friday), collected from Datastream.4 The oil price (OIL WTI) is the price of West Texas Intermediate (WTI), expressed in U.S.$/barrel. ret OIL WTI represents the first difference of log of (consecutive) Friday prices of OIL WTI. For foreign exchange market, EX MAJ is a weighted average of the foreign exchange value of the U.S. dollar against a subset of the major index currencies including the Euro Area, Canada, Japan, United Kingdom, Switzerland, Australia, and Sweden. The index has base Mar 1973 = 100 and an increase means a USD appreciation. ret EX MAJ is the first difference of log of (consecutive) Friday values of EX MAJ index. Two measures of ex-ante real interest rate are computed using the formula below: real interestt = FFRt − (treasuryt − TIPSt )

(1)

where real interestt is the ex-ante real interest rate at time t, FFRt is the target federal funds rate at time t, treasuryt and TIPSt are the yield of 5-year U.S. T-note and yield of 5-year U.S. Treasury Inflation Protected security at time t, respectively.5 Mollick and Assefa (2013) adopt a similar measure when studying U.S. stock markets under GARCH methods. The term (treasuryt − TIPSt ) is also called expected inflation at time t and is plotted in graph of Fig. 1. Inflation expectations have mean of 2.12% and 1.73%, depending on the time period, as Table 1 indicates. It is plausible to assume a level slightly different from 2% for inflation expectations in the U.S., except for the sharp drop during the 2008 recession when actual inflation became negative temporarily. Fig. 1 also indicates that the real interest rate is far from constant. We identify two subsamples to test changes in the relationships among the series over time by picking up the date on which real interestt becomes negative. According to Fig. 1, there is a

3 The first version of this paper used daily data from July 10, 1997 to September 13, 2013. As suggested by an anonymous referee, daily data often incorporate too much noise and we shift to the use of weekly data. The results of daily data version are similar to those in this version and are available upon request. However, under weekly data, we attempt to have symmetry in the two time intervals in order to highlight as much as possible the effects of the recent financial crisis and quantitative easing on nominal (and real) interest rates and thus on the stock market. In our original daily data analysis the number of observations in the first subperiod was 3014 and 1208 in the second subperiod. In the current weekly data, the number of observations in the first subperiod is 317 and 322 in the second subperiod, which make the subsamples more comparable for the sake of stock markets effects. 4 Following Chordia and Swaminathan (2000), we use every Friday prices to obtain weekly returns or differences. 5 Our work on daily data also used yields in 10-year U.S. T-note and TIPS and the results were similar.

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period before the 2008–2009 when real interestt drops below zero. However, that was short-lived, recovering afterwards to the more normal positive rate. It is clear from Fig. 1 that, following the severity of the crisis, real interest rate became negative and remained negative until the time of writing. We therefore split the data right at the most recent point when real interestt crossed below zero: on January 29, 2009, since real interestt remains negative throughout after that date.6 There are, of course, many ways to identify a change in monetary policy regime. The approach herein emphasizes the sustained amount of time interest rates become very close to zero, which will push up stock prices in the discounted cash flow framework by Chen et al. (1986). Focusing, however, on the real interest rates we combine nominal interest rates (that discount expected earnings) and price expectations in one single measure. This is the (ex-ante) real interest rate, which can be interpreted theoretically as the marginal product of capital.7 Previous studies for U.S. stocks have attempted to gauge these forces which appear confounded by effects from interest rates and from expected cash flows. Campbell and Ammer (1993) use monthly postwar data in VARs to show that real interest rate changes have little impact on either stock or 10-year bond returns, although they do affect the short-term nominal interest rate and the slope of the term structure. The theoretical framework starts with the log real return on a stock, ht + 1 , defined by ht + 1 = log(Pt + 1 − Dt + 1 ) − log(Pt ), where P and D are the levels of the end-of-period real stock price and dividend, respectively. A firstorder Taylor approximation of this equation can be thought as a difference equation relating (dt − pt ) to (dt + 1 − pt + 1 ), dt + 1 and hj + 1 . Solving forward, and imposing the transversality condition, the log dividend-price ratio (dt − pt ) can be written as a discounted value of all future returns and dividend growth rates discounted at a constant rate. Additional VARs in Thorbecke (1997) for U.S. monthly data from 1967 to 1990 for four different measures of monetary policy indicate that expansionary (contractionary) monetary policy causes stock returns (of portfolios) to increase (decrease). His interpretation for this robustness was that the findings are not merely capturing an interest rate effect. Table 1 shows descriptive statistics of variables used in this study both in levels and in returns/differences. Panel A reports for the period on or before January 28, 2009 and Panel B reports for the period after January 28, 2009. According to Table 1A, the mean return of oil price (0.0010) is relatively higher than those of exchange rate (−0.006) or 3 stock market indices returns (varying from −0.0003 in S&P 500 to 0.0002 in NASDAQ) for the first period studied. However, in the second period, the mean return of oil price (0.0002) is relatively smaller than those of exchange rate (0.0003) or 3 stock market indices returns (varying from 0.0024 in Dow Jones to 0.0037 in NASDAQ). The Sharpe ratio indicator (mean divided by

6 Cecchetti (2009) surveys the critical period in monetary policy and identifies the policy actions taken by the FED between Aug 9, 2007 and May 2, 2008, into two groups: those that fit the conventional textbook definitions of aggressive monetary policy and those that do not. The first group comprises the seven cuts in the target federal funds rate (FFR) totaling 3.25% from Sep 18, 2007 to Apr 30, 2008. The second group includes a series of less conventional actions. See Cecchetti (2009, p. 65) for a table with conventional and unconventional actions. 7 We could alternatively have split our sample at September 18, 2007 (50 bp cut at regular scheduled FOMC meeting), which would indicate an aggressive change in policy to the downside. Adopting this alternative cutoff, the qualitative results are unchanged and we prefer to focus on the prolonged period of negative real interest rates since this makes our argument closer to the theoretical link of very low interest rates making stock prices increase. Checking the federal fund rate from the St. Louis FED, the federal fund target rate was 5.33% on September 17, 2007; on September 18 it was 4.92%; and on September 19 it was 4.74%, after the half-full point cut. The results for this alternative cutoff (with Sep 18, 2007 indicating a clear change in policy) under daily data are available upon request from the authors.

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Table 1 Descriptive statistics (levels and returns/differences). Mean

Median

Min

Max

Std. dev.

Panel A: Period on or before January 28, 2009 61.06 59.13 OIL WTI ret OIL WTI 0.0010 0.0067 EX MAJ 82.91 82.58 −0.0006 −0.0018 ret EX MAJ 0.0371 0.0368 Treasury 5yr 0.0159 0.0142 TIPS 5yr 0.0212 0.0231 expected inflation FFR 0.0292 0.0250 real interest 0.0081 0.0057 diff real interest 0.0000 0.0000 10,916.02 10,678.56 DJINDUSPI −0.0002 0.0018 ret DJINDUSPI 2115.41 2135.08 NASCOMPPI ret NASCOMPPI 0.0002 0.0020 1214.08 1222.12 SPCOMPPI −0.0003 0.0016 ret SPCOMPPI

25.74 −0.3573 69.68 −0.0382 0.0135 0.0008 −0.0224 0.0025 −0.0160 −0.0212 7740.03 −0.2003 1282.47 −0.1660 800.03 −0.2008

145.31 0.2432 100.45 0.0435 0.0521 0.0417 0.0292 0.0525 0.0329 0.0084 14,093.08 0.1070 2810.38 0.1037 1561.80 0.1136

Panel B: Period after January 28, 2009 85.89 OIL WTI ret OIL WTI 0.0002 75.83 EX MAJ 0.0003 ret EX MAJ 0.0153 Treasury 5yr −0.0020 TIPS 5yr expected inflation 0.0173 0.0025 FFR real interest −0.0148 0.0000 diff real interest 12,963.73 DJINDUSPI 0.0024 ret DJINDUSPI 3049.34 NASCOMPPI ret NASCOMPPI 0.0037 1418.09 SPCOMPPI ret SPCOMPPI 0.0028

37.63 −0.1575 68.19 −0.0385 0.0059 −0.0167 0.0037 0.0025 −0.0220 −0.0036 6626.94 −0.0662 1293.85 −0.0847 683.38 −0.0746

113.39 0.1267 93.37 0.0270 0.0285 0.0147 0.0245 0.0025 −0.0012 0.0035 18,140.44 0.0863 5026.42 0.1011 2110.30 0.1017

89.94 0.0022 75.39 0.0001 0.0156 −0.0019 0.0179 0.0025 −0.0154 −0.0001 12,700.82 0.0041 2866.31 0.0044 1343.41 0.0035

Skewness

Kurtosis

Obs.

Sharpe ratio

26.07 0.0619 6.67 0.0110 0.0085 0.0067 0.0073 0.0169 0.0153 0.0019 1493.43 0.0244 343.91 0.0290 184.14 0.0252

1.04 −1.1609 0.17 0.5881 −0.2896 0.5781 −3.0312 0.1891 0.1687 −4.3358 0.07 −1.5064 −0.42 −0.8611 −0.23 −1.5415

3.82 9.3221 3.02 4.3141 2.4879 3.4565 13.9473 1.4588 1.5467 52.1751 2.42 18.3395 2.72 7.6068 2.36 17.6529

317 316 317 316 317 317 317 317 317 316 317 316 317 316 317 316

– 0.0156 – – – – – – – – – −0.0082 – 0.0069 – −0.0111

16.59 0.0418 4.40 0.0094 0.0059 0.0077 0.0036 0.0000 0.0036 0.0010 2807.47 0.0215 899.24 0.0253 351.26 0.0233

−0.90 −0.3949 1.34 −0.0439 0.1416 0.1236 −1.1258 . 1.1258 −0.1621 0.08 −0.0807 0.44 −0.1551 0.34 −0.1306

3.23 4.4369 5.44 3.9383 2.0693 2.3031 4.7899 . 4.7899 4.6269 2.09 4.6378 2.27 4.5476 2.13 4.9142

322 322 322 322 322 322 322 322 322 322 322 322 322 322 322 322

– 0.0059 – – – – – – – – – 0.1136 – 0.1469 – 0.1210

Notes: This table presents descriptive statistics for weekly data used in this study. Panel A reports descriptive statistics for period on or before January 28, 2009 and Panel B reports descriptive statistics for period after January 28, 2009. The oil prices, OIL WTI, are prices of West Texas Intermediate (WTI), expressed in US dollars per barrel. EX MAJ is a weighted average of the foreign exchange value of the U.S. dollar against a subset of the major index currencies. ret OIL WTI and ret EX MAJ represent first differences of log of (consecutive) Friday prices of OIL WTI and EX MAJ, respectively. Treasury 5yr and TIPS 5yr are the yield of 5-year U.S. T-note and yield of 5-year U.S Treasury Inflation Protected security, respectively. expected inflation is the expected inflation. FFR is the target federal funds rate. real interest is the ex-ante real interest rate computed using the method in section 2 and diff real interest is consecutive weekly differences real interest (using the rate of every Fridays). DJINDUSPI, NASCOMPPI and SPCOMPPI are every Friday index prices for Dow Jones, NASDAQ and S&P 500 market indices. ret DJINDUSPI, ret NASCOMPPI and ret SPCOMPPI represent first differences of log of (consecutive) Friday prices of DJINDUSPI, NASCOMPPI and SPCOMPPI, respectively. Sharpe ratio indicates the isk-adjusted performance of returns and is calculated by dividing the mean return by its associate standard deviation.

standard deviation) indicates that crude oil price has higher returnadjusted-for-risk than the three major U.S. stock markets only in the earlier period. In the second, stock markets have higher ratios of return adjusted by risk (varying from 0.1136 in Dow Jones to 0.1469 in NASDAQ) than crude oil (0.0059). Crude oil returns also have highest standard deviation among other return series for both periods studied. This is consistent with documented evidence on the comparatively high volatility of commodities. The value of the U.S. dollar against major currencies (ret EX MAJ) shows a negative return of -0.0006% for the period before January 28, 2009, consistent with the U.S. Dollar weakening in the crisis period (as the real return on U.S. assets is negative). The real interest rate shows of course a very different mean for the two periods studied: While mean real interest rate in the earlier period is 0.0081, the figure for the 2009–2015 period is −0.0148, a change of more than 2%, reflecting the accommodative U.S. monetary policy in the more recent period. For the three major U.S. stock market indices, their mean returns are close to zero in the period on or before January 28, 2009. In the more recent period, those figures are well above zero, suggesting broad gains in the U.S. stock markets. More recently, NASDAQ experiences the highest mean return of 0.37% and Dow Jones experiences the lowest mean return of 0.24%, with the S&P mean return in between. Fig. 2 contains several graphs that help visualize the relationships between the time series. The graph at the upper-left of Fig. 2

suggests positive co-movements between S&P 500 index and the oil price for most of the period. In the graph at middle-left of Fig. 2, S&P 500 seems to have positive relationship with real interest rate in the period on or before January 28, 2009. After January 28, 2009 when real interest rate becomes negative and flat S&P 500 index continuously moves higher. We use an unrestricted VAR model to investigate the interactions among the aforementioned series, following the dynamic interactions among these four financial markets.8 A VAR model has been frequently used to analyze the impact of oil price shocks on other economic and financial series. Examples include Sadorsky (1999), Huang, Hwang, and Peng (2005), Kilian and Park (2009), and Lee, Yang, and Huang (2012). Our VAR model is estimated as follows:

yt = B0 +

p 

Bi yt−i + ε1t

(2)

i=1

8 As emphasized by Engle and Granger (1987), variables with cointegration should not be used in the VAR model. Using the bounds testing method developed by Pesaran et al. (2001) to test whether there is cointegration among the series, the results are supportive of no cointegration and we focus on short-run VAR responses in this paper.

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Fig. 2. Movements of equity market (S&P 500), oil price, ex-ante real interest rates and U.S. dollar index.

where yt is vector of the four series studied in this paper (ret OIL WTI, ret EX MAJ, diff real interest and one of the following stock markets: ret DJINDUSPI, ret NASCOMPPI, or ret SPCOMPPI). B0 is a 4 × 1 column vector of constant terms, Bi is a 4 × 4 matrix of coefficients and ε1t is a 4 × 1 column vector of error terms. The specific ordering used runs from the most exogenous market (oil) to the most endogenous (stock markets), with exchange rates and real interest rates in between. Variance decompositions will shed light on the plausibility of this particular ordering and the use of generalized impulse response functions (GIRFs) allows responses of variables to shocks of other variables which are robust to the particular ordering of variables in the VAR system. Note also that since all series are used in returns or in firstdifferences, the vector is covariance stationary, or first-order weakly stationary. The number of lags determined in (2) will be based on Akaike Information Criterion and Final Prediction Error as proposed by Hamilton (1994). From the VAR model, we extract the forecast error variance decompositions (using the Monte Carlo method with 5000 replications), as well as the generalized impulse response functions (also with 5000 replications). The forecast error variance decompositions show how much of the variance of a variable can be explained by shocks to another variable in the model. In order to identify the dependence pattern among the commodity and exchange rate markets, monetary policy, and stock market, the vine copula methodology is applied. Copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval (0, 1). A vine is a flexible graphical model for

describing multivariate copulas built up using a cascade of bivariate copulas, so-called pair-copulas. A regular vine (R-vine) is a special case of a vine, and canonical (C-) and drawable (D-) vines are two special cases of an R-vine. The definitions of vine, R-vine, C-vine and D-vine can be found in Bedford and Cooke (2002), the statistical technique of C-vine and D-vine can be found in Aas et al. (2009), and that of R-vine can be found in Dißmann et al. (2013). Maximum likelihood estimation method is used in vine copula model fitting, which requires that there should be no autocorrelations in each data series. After testing for autocorrelations by the Ljung–Box test, we use an AR(1)-GARCH(1,1) model to filter our data. Following the selecting and estimating procedure introduced in Dißmann et al. (2013), R-vine copula structures for our data are found. Since C-vine is widely used in the literature, in the case that the found R-vine is not a C-vine, we also use C-vine copula to fit the data and then use a Vuong test (Vuong, 1989) to compare the two models. If the test shows that the two models are not statistically different, we use the C-vine. Finally, we apply the goodness-of-fit test proposed by Schepsmeier (2013) to verify that the selected vine copula model is appropriate. 3. Empirical results 3.1. Correlation coefficients Tables 2A and 2B report the correlation matrices in levels and in returns/differences, respectively. The results from Table 2A are

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Table 2A Correlation matrix (levels). OIL WTI Panel A: Period on or before January 28, 2009 1.0000 OIL WTI −0.8527 EX MAJ real interest 0.3388 0.6411 NASCOMPPI DJINDUSPI 0.6388 0.6184 SPCOMPPI Panel B: Period after January 28, 2009 OIL WTI 1.0000 −0.6391 EX MAJ real interest −0.7703 0.3854 NASCOMPPI 0.2972 DJINDUSPI 0.3260 SPCOMPPI

EX MAJ

real interest

NASCOMPPI

DJINDUSPI

SPCOMPPI

1.0000 −0.3708 −0.7610 −0.7501 −0.7166

1.0000 0.5967 0.5730 0.6414

1.0000 0.9688 0.9835

1.0000 0.9792

1.0000

1.0000 0.5915 0.3177 0.3992 0.3762

1.0000 −0.3363 −0.2448 −0.2788

1.0000 0.9883 0.9944

1.0000 0.9968

1.0000

Notes: This table presents correlation coefficients for weekly data series, in their level, used in this study. Panel A reports descriptive statistics for period on or before January 28, 2009 and Panel B reports descriptive statistics for period after January 28, 2009. Description of variables is described in Section 2 and in Table 1’s notes.

Table 2B Correlation matrix (differences/returns). ret EX MAJ

diff real interest

ret DJINDUSPI

ret NASCOMPPI

ret SPCOMPPI

Panel A: Period on or before January 28, 2009 1.0000 ret OIL WTI −0.3119 ret EX MAJ 0.1154 diff real interest 0.0979 ret NASCOMPPI 0.0903 ret DJINDUSPI ret SPCOMPPI 0.1407

ret OIL WTI

1.0000 0.1179 −0.2277 −0.2466 −0.2806

1.0000 −0.1379 −0.1449 −0.1605

1.0000 0.8809 0.9744

1.0000 0.9243

1.0000

Panel B: Period after January 28, 2009 1.0000 ret OIL WTI −0.3952 ret EX MAJ diff real interest −0.4772 ret NASCOMPPI 0.3925 ret DJINDUSPI 0.3802 0.4206 ret SPCOMPPI

1.0000 0.3085 −0.3869 −0.3690 −0.4067

1.0000 −0.5301 −0.5066 −0.5399

1.0000 0.9023 0.9770

1.0000 0.9491

1.0000

Notes: This table presents correlation coefficients for weekly data series, in their difference or return, used in this study. Panel A reports descriptive statistics for period on or before January 28, 2009 and Panel B reports descriptive statistics for period after January 28, 2009. Description of variables is described in section 2 and in Table 1’s notes.

generally consistent with the graphic patterns in Fig. 2. According to Table 2A, oil price shows positive correlations with all major U.S. stock market indices and these positive correlations are much larger (and similar across stock markets) in the first period with 0.6411 for NASDAQ, 0.6388 for Dow Jones and 0.6184 for S&P 500. But the correlation between oil and stocks is higher in the second period when return is used instead of raw data. Negative correlations are examined between exchange rate index and stock markets in the period on or before January 28, 2009. These negative correlations are more profound when level data are used. However, the correlations between exchange rate index and stock markets at the period after January 28, 2009 yield interesting results. While there are positive correlations (EX MAJDJINDUSPI shows the largest figure of 0.3992) when level data are used, there are negative correlations (ret EX MAJ-ret SPCOMPPI shows the largest absolute figure of −0.4067) when return data are used. The correlations between real interest rate and stock markets at the period on or before January 28, 2009 yield very different results. In levels, stocks and real interest rate move inversely only in the recent period, consistent with the notion that stock returns surge when real interest rate declines. For the recent subsample, the correlations between real interest rate and stock markets are clearly negative: between −0.2448 and −0.3363 in levels or between −0.5066 and −0.5399 in differences/returns.

3.2. Variance decompositions and impulse response functions Table 3 reports – for a 5-week forecasted horizon period – the variance decomposition of the VAR models in (2) for S&P 500.9 Based on information criteria, we employ p = 9 lags for the VAR in the first sub-period and p = 1 lags for the VAR in the second sub-period.10 Table 3 Panel A reports the VAR model with S&P 500 as U.S. stock market benchmark for the period on or before January 28, 2009 while Table 3 Panel B reports the results for the period after January 28, 2009. In Panel A, the variance of oil price returns, exchange rate index returns, difference in real interest rate and S&P 500 returns are significantly explained only by their own shocks. At week 5, shocks in oil price returns explain 88.10% of the variance of oil price returns. Shocks in exchange rate index returns explain 81.81% of

9 For robustness check, we also replicate our VAR analysis using Dow Jones and NASDAQ indices in exchange for S&P 500 index. These results are similar and available upon request. 10 Under this choice of lag-length, application of VAR residual serial correlation LM tests at 5 lags suggest good properties in general for all markets as all markets passed the null of no serial correlation for the tests using 5 lags. For the S&P 500, for example, we do not reject the null using 1% level since LM-stat = 19.5053 (p-value of 0.2433) in the first sub-period and LM-stat = 9.7695 (p-value of 0.8784) in the more recent period.

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Table 3 Variance decomposition of VAR model for S&P 500. Variance decompositions across weeks

Panel A: Period on or before January 28, 2009 RET OIL WTI 1 2 3 4 5 RET EX MAJ 1 2 3 4 5 DIFF REAL INTEREST 1 2 3 4 5 RET SPCOMPPI 1 2 3 4 5 Panel B: Period after January 28, 2009 RET OIL WTI 1 2 3 4 5 RET EX MAJ 1 2 3 4 5 DIFF REAL INTEREST 1 2 3 4 5 RET SPCOMPPI 1 2 3 4 5

Innovation Shock in

Shock in

Shock in

Shock in

RET OIL WTI

RET EX MAJ

DIFF REAL INTEREST

RET SPCOMPPI

100.00 98.79 95.21 92.66 88.10

(0.00) (1.05) (2.38) (3.43) (4.61)

0.00 0.38 0.29 1.18 2.23

(0.00) (0.62) (0.72) (1.54) (2.35)

0.00 0.28 0.29 0.52 1.97

(0.00) (0.50) (0.66) (1.07) (1.97)

0.00 0.55 4.21 5.63 7.70

(0.00) (0.69) (2.20) (2.99) (3.80)

13.70 13.39 13.32 13.51 13.56

(3.62) (3.55) (3.48) (3.49) (3.49)

86.30 84.22 82.25 81.94 81.81

(3.62) (3.86) (3.99) (3.99) (4.02)

0.00 2.09 2.64 2.70 2.73

(0.00) (1.70) (1.86) (1.87) (1.90)

0.00 0.30 1.79 1.85 1.90

(0.00) (0.78) (1.57) (1.64) (1.71)

0.65 3.53 3.69 3.64 3.63

(0.99) (2.21) (2.25) (2.24) (2.25)

1.71 2.14 2.58 2.56 2.56

(1.51) (1.74) (1.90) (1.92) (1.97)

97.64 93.18 92.51 91.19 90.94

(1.78) (2.96) (3.10) (3.31) (3.37)

0.00 1.15 1.22 2.61 2.87

(0.00) (1.29) (1.39) (1.90) (2.00)

2.00 2.01 3.96 4.30 4.80

(1.64) (1.71) (2.16) (2.25) (2.39)

6.87 6.83 6.69 6.85 6.71

(2.74) (2.76) (2.61) (2.62) (2.60)

2.04 2.78 4.89 5.05 6.41

(1.57) (1.93) (2.61) (2.73) (3.05)

89.09 88.39 84.46 83.80 82.09

(3.37) (3.56) (3.93) (4.02) (4.18)

100.00 99.73 99.70 99.68 99.66

(0.00) (0.54) (0.62) (0.68) (0.72)

0.00 0.03 0.04 0.04 0.04

(0.00) (0.25) (0.33) (0.37) (0.40)

0.00 0.22 0.24 0.26 0.28

(0.00) (0.40) (0.44) (0.48) (0.50)

0.00 0.02 0.02 0.02 0.02

(0.00) (0.25) (0.29) (0.32) (0.34)

14.63 14.80 14.81 14.81 14.81

(3.67) (3.64) (3.64) (3.64) (3.64)

85.37 84.64 84.63 84.63 84.63

(3.67) (3.77) (3.77) (3.77) (3.77)

0.00 0.18 0.18 0.18 0.18

(0.00) (0.54) (0.55) (0.55) (0.55)

0.00 0.38 0.38 0.38 0.38

(0.00) (0.78) (0.78) (0.78) (0.78)

22.26 21.76 21.75 21.80 21.84

(4.03) (3.96) (3.96) (3.95) (3.95)

2.04 3.20 3.21 3.21 3.21

(1.43) (1.70) (1.70) (1.70) (1.70)

75.70 75.04 75.03 74.98 74.94

(4.08) (4.05) (4.04) (4.04) (4.04)

0.00 0.00 0.01 0.01 0.01

(0.00) (0.42) (0.44) (0.44) (0.44)

18.45 18.37 18.37 18.38 18.38

(3.91) (3.88) (3.87) (3.87) (3.87)

7.51 7.67 7.67 7.67 7.67

(2.57) (2.58) (2.58) (2.58) (2.58)

11.65 11.74 11.74 11.74 11.74

(2.89) (2.90) (2.90) (2.90) (2.90)

62.39 62.22 62.22 62.21 62.21

(4.27) (4.26) (4.26) (4.26) (4.26)

Notes: This table reports the variance decomposition of 2 VAR models, before (Panel A) and after (Panel B) January 28, 2009, for ret OIL WTI, ret EX MAJ, diff real interest and ret SPCOMPPI using Monte Carlo method with 5000 repetitions. Numbers of lag-length used in these VAR models are 9 and 1 for period before and after January 28, 2009, respectively, as indicated by Final Prediction Error and Akaike Information Criterion in VAR lag order selection procedure. Standard errors are reported in parenthesis.

the variance of exchange rate index returns. Shocks in real interest rate difference explain 90.94% of the variance of real interest rate difference and shocks in S&P 500 returns explain 82.09% of the variance of S&P 500 returns. One implication from Panel A of Table 3 is that, before January 28, 2009, the relationships among commodity, exchange rate, monetary policy and stock markets are weak. Table 3 Panel B yields very interesting results. In the period after January 28, 2009, while the variance of oil price returns is still solely explained by shocks of itself the variances of exchange rate index returns, real interest rate differences and S&P 500 returns are well explained by shocks of other variables. Shocks in oil price returns

are able to explain 21.84% of the variance of real interest rate difference. In addition, shocks in oil price returns and real interest rate differences are able to explain, respectively, 18.38% and 11.74% of the variance of S&P 500 returns. All these are statistically significant. These results imply that, after January 28, 2009, the linkages among commodity and exchange rate markets, monetary policy, and stock market are much stronger than before. Fig. 3 reports the selected impulse response of VAR models for S&P 500 for the 5-week forecasted period. The responses to generalized one standard deviation innovations ±2 standard error (confidence bands) are plotted in the y-axis and time (weeks) is

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Fig. 3. Impulse responses of VAR model using S&P500 index as stock market (using weekly data). Panel A: Period on or before January 28, 2009. Panel B: Period after January 28, 2009. Notes: Using weekly data, this figure reports impulse response function of VAR models before (Panel A) and after (Panel B) January 28, 2009 for ret OIL WTI, ret EX MAJ, diff real interest and ret SPCOMPPI. The VAR model uses Monte Carlo method with 5000 repetitions, the responses to generalized one standard deviation innovations ±2 standard error (confidence bands) are plotted in the y-axis and time (weeks) is plotted in the x-axis.

plotted in the x-axis. We will focus on the responses of real interest rates and stock markets since oil and exchange rates are relatively more exogenous, being determined in international markets. Indeed, the variance decompositions in Table 3 clearly confirm this for oil and also, but by a lesser extent, for foreign exchange.11 Due to space constraints, we interpret in detail the VAR model using S&P 500 index as stock market index. According to Fig. 3, real interest rate reacts negatively to the shock of oil price and the significance level and the magnitude of the reaction is more profound in the period after January 28, 2009. One way to rationalize this

11 In the VAR analysis using Dow Jones index as the stock market (available upon request), for example, oil returns respond to their own shocks by 89.77% (first period) and 99.68% (second period) after 5 weeks. For the NASDAQ stock market, oil returns respond to their own shocks by 90.01% (first period) and 99.66% (second period) after 5 weeks. For the S&P 500 stock market, oil returns respond to their own shocks by 88.10% (first period) and 99.66% (second period) after 5 weeks. As for the value of the U.S. dollar against major currencies, the high percentage of exchange rate returns responding to their own shocks – after 5 weeks – is also very high in the first period in all cases: 81.74% (Dow), 81.71% (NASDAQ), and 81.81% (S&P 500), after 5 weeks. But these levels increase in the more recent period to 84.40% (Dow), 85.00% (NASDAQ), and 84.63% (S&P 500). Overall, returns in FX respond mostly to their own shocks, although at lower levels than oil price returns.

response is looking at oil as a major input resource. If input costs rise, prices of final products will rise, leading to a higher inflation rate. Since the real interest rate is constructed by the nominal interest rate minus the inflation rate, high inflation will move the real interest rate downwards. For example, in the period on or before January 28, 2009 the 1% increase in oil price shock leads to −0.0128 basis points decrease in the real interest rate. However, this impact is not significant as shown by the confidence bands in Fig. 3B. The same figure for the period after January 28, 2009 is statistically significant and almost four times that level before that at −0.0472. This is the point estimate at the time of the shock following a one percent increase in the shock. In both subsamples, this effect on real interest rate will disappear after 2 weeks for both periods, consistent with the fast rate of assimilation of information in financial markets. Looking at the relationship between real interest rate and the U.S. dollar in the second column, there is a small significant effect of a shock to U.S. dollar value on real interest rate in the period on or before January 2009 while there is a positive effect in the following period. In response to a one standard deviation in the value of the USD, there are increases in the real interest rate in both periods, with a close to 0.03% positive response in the second subsample.

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Fig. 4. Plots of pairwise dependency. Notes: This figure shows pairs plot with scatter plots above the diagonal and contour plots below. Axes of the contour plots range from −3 to 3 other than indicated here.

An increase in shocks to the USD leads to a higher real price of U.S. dollar-based assets, which is intuitive. On the relationship between oil price and stock markets, there is a small effect between shocks of oil prices and S&P 500 index returns in the period on or before January 28, 2009 while there is a positive and significant effect in the second period. Specifically, after January 28, 2009, the 1% increase in oil price shock will lead to 0.01% increase in S&P 500 index return. This is consistent with GARCH models by Mollick and Assefa (2013), who argue that oil prices contain expectations of world economic recovery and impact stocks positively after the U.S. downturn of 2008–2009. Also, stock markets respond negatively to positive shocks to the value of the USD (consistent with a trade-based channel) in the second sub-period: a stronger USD lead to higher trade deficits and thus to lower GDP growth, which affects negatively equity prices. The response is still positive but very close to zero in the first sub-period. One would expect a negative relationship between real interest rate and the value of stock market since the present value of stock equals to sum of all expected cash-flows discounted by the interest rate. The response of S&P 500 stock returns is negative and statistically significant but small in the first sub-period. It is, however, negative and significantly large in the second sub-period: in response to an increase of 1% in the standard deviation of real

interest rates, stocks fall close to 1.26%. The results from Fig. 3 not only do support this hypothesis from the expected cash flow model but also show a deeper effect of shocks of real interest rate changes on the return of S&P 500 index under the expansionary monetary policy performed by the U.S. Federal Reserve in the more recent years. 3.3. Vine copula modeling The above analysis shows that, after January 28, 2009, the linkages among commodity and exchange rate markets, monetary policy, and stock market are much stronger than before. An interesting question is: What is the dependence pattern among commodity and exchange rate markets, monetary policy, and stock market? In order to answer this question, we apply the vine copula methodology. In the model fitting, we need to use the maximum likelihood estimation method which requires that there should be no autocorrelations in each data series. It is well-known that time series typically exhibit autocorrelations. A Ljung–Box test indicates that, except for exchange rate returns, all other five series in our data set exhibit autocorrelations. Thus, a univariate time series model is necessary to filter the data. It would be desirable if we would have all series in log-difference form which would guarantee that

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the data used for univariate time series modeling is covariancestationary. A difficulty is that we cannot take logarithm for the real interest rate series because it has many negative observations. One solution is to add a very small constant to all its observations to make them positive. This is based on the fact that adding a constant to a series does not change either the linear correlation or the rank correlation of this series with any other ones. An alternative way to make the series covariance-stationary is to take the growth rate of the series, which can be shown to be approximately the same as log-difference. See, e.g., Table 3.1 of Gonzalez-Rivera (2013). We use the latter to obtain covariance-stationarity.12 Our main purpose is to model the dependence among the series. Therefore, the dependence should not be changed after univariate filtering. In order to not affect the relative dependence among series, one filter should be chosen for all series. Although series such as exchange rate return does not exhibit autocorrelations, it should still be filtered using the same model; otherwise the dependence among series would be changed. It turns out that, while a skewed normal AR (1)-GARCH (1,1) model is appropriate to filter all series before January 28, 2009, we need to use a skewed t AR (1)-GARCH (1,1) model to filter series after January 28, 2009. This suggests there are fat tails in series after January 28, 2009. A Ljung–Box test verifies that there are no autocorrelations among the standardized residuals, from which the copula data are obtained. For a first impression of the pairwise dependence, Fig. 4 presents a pairs plot with scatter plots above the diagonal and contour plots below. Fig. 4 shows that there exist both symmetric and asymmetric tail dependences between series: some dependences are positive and others are negative. The dependences between returns of exchange rate and stocks are all negative, which support the correlation coefficients of Table 2B and the impulse responses of the VAR models. The strongest dependence is between the stock return pairs, which could be an interesting topic for future research in order to shed light on the causation patterns of different equity markets at high frequencies. In order to capture the co-movement among series, we need to find the vine copula structure and estimate the pair-copula parameters. Using the selecting and estimating procedure introduced in Dißmann et al. (2013), the vine copula structures among various series are found. Fig. 5 presents the vine copula structure for all six series, i.e., the vine copula structure for the multivariate distribution of all three stock returns with oil price return, exchange rate return and real interest rate return. The vine copula structures for Dow Jones return and NASDAQ return with oil price return, exchange rate return and real interest rate return are found to be only slightly different from that for S&P 500 return with the three series. In order to save space, Fig. 6 presents only the vine copula structure for S&P 500 return with oil price return, exchange rate return and real interest rate return. It turns out that all vine copula structures above are C-vines. A typical property of C-vines is that there is a central variable (or pair) in each tree of the vine structure, and the relationships of this variable (or pair) with each of other variables (or pairs) capture the dependence in a specific tree. It is well-known that, in a vine structure, the most important (and strongest) dependencies are modeled in the first trees (see, e.g., Brechmann & Joe, 2015; Brechmann, Czado, & Aas, 2012). Both Figs. 5 and 6 show that S&P 500 return is in the center of the first tree which shows how S&P 500 return moves along with oil price return, exchange rate return and real interest rate return, and the bivariate relation of S&P 500 return with oil price return is in the center of the second tree

12 This is inspired by the comments from one of the referees. The authors wish to say thanks.

Fig. 5. Vine copula structure (6 series). Notes: Vine copula structure for the multivariate distribution of all three stock returns (ret DJINDUSPI) with oil price return (ret OIL WTI), exchange rate return (ret EX MAJ) and real interest rate return (ret real interest). The numbers in the figure represent different series: 1 for oil price return, 2 for exchange rate return, 3 for real interest rate return, 4 for Dow Jones index return, 5 for NASDAQ index return, and 6 for S&P 500 index return.

Fig. 6. Vine copula structure (4 series). Notes: Vine copula structure for the multivariate distribution of S&P 500 return (ret SPCOMPPI) with oil price return (ret OIL WTI), exchange rate return (ret EX MAJ) and real interest rate return (ret real interest). The numbers in the figure represent different series: 1 for oil price return, 2 for exchange rate return, 3 for real interest rate return and 4 for S&P 500 index return.

which suggests that the co-movement of S&P 500 return with oil price return is a leading force. Since the results are similar for all stock returns, we focus on the S&P 500 only, as before. As mentioned before, the copula evidence is for the second subsample only. Table 4 presents the copula families for each pair-copula and the corresponding estimation results for pair-copula parameters corresponding to the vine structure in Fig. 6. Table 4 indicates that, as predicted in Fig. 4, there are both positive and negative dependences, and tail dependences could be either symmetric or asymmetric. As mentioned above, in a vine structure, the most important (and strongest) dependencies are modeled in the first trees. Looking at the first tree only,

W. Huang et al. / The Quarterly Review of Economics and Finance 59 (2016) 231–242 Table 4 Vine copula estimation results. Pair-copula

Copula family

14 24 34 12|4 13|4 23|14

Clayton t Gaussian Gaussian Gaussian Frank

Copula parameter (s.e.) 0.6143 (0.0934) −0.3813 (0.0545) 0.5325 (0.0356) −0.2620 (0.0516) 0.3387 (0.0486) −0.3855 (0.3335)

Degrees of freedom (s.e.) 4.0030 (1.2192)

Notes: This table reports vine copula estimation results for the multivariate distribution of S&P 500 return with oil price return, exchange rate return and real interest rate return. The numbers in each pair-copula represent different series: 1 for oil price return (ret OIL WTI), 2 for exchange rate return (ret EX MAJ), 3 for real interest rate return (ret real interest) and 4 for S&P 500 return (ret SPCOMPPI). Details of pair-copula are reported in Fig. 6 or Section 3.3. Standard errors are reported in parenthesis.

there is more dependence for oil and stocks, followed by real interest rate and stocks, and then currency and stocks. However, for the first tree, all 3 pairs are from different families: Clayton, Gaussian and t, respectively. Specifically, Clayton copula parameter 0.6143 suggests positive asymmetric tail dependence between the oil price return and the S&P 500 return, while t copula parameter −0.3813 suggests negative symmetric tail dependence between the exchange rate return and the S&P 500 return. The parameter of the bivariate Gaussian copula between real interest rate return and the S&P 500 return is 0.5325. One interpretation is that, in the very low rate environment of second subsample (mean of real interest rate is −0.0148 from Table 1), dependence between stock returns and real interest rate return is positive. We present in Table 4 only the results for S&P 500 return to save space, with results for Dow Jones and NASDAQ available upon request. It turns out that there is tail dependence between Dow Jones return and real interest rate return, but no tail dependence between other two stock returns and real interest rate return. Now that we have fitted C-vines to the dependence among stock return, oil prices, exchange rates and U.S. real interest rates, one may be wondering whether our vine copula modeling is a good fit. In this regard, Schepsmeier (2013) extends the goodness-offit test for copulas introduced by Huang and Prokhorov (2014) and develops a goodness-of-fit test for regular vine copula models. Since our sample size is finite, we apply this method bootstrapped 200 times to test whether the C-vine detailed in Fig. 6 and Table 4 is appropriate (see, e.g., Schepsmeier, 2015). The testing results, not reported in Table 4 but available upon request, show that the White test statistic is 23.8734 and the p-value is 1, which suggests that our vine copula modeling is a very good representation. 4. Concluding remarks This paper addresses the response of U.S. stock markets to fluctuations in oil prices, exchange rates, and in U.S. real interest rates using weekly data from January 3, 2003 to March 27, 2015. We report higher correlation coefficients and magnified impulse responses and variance decompositions when real interest rates become negative in early 2009. U.S. stock markets respond – in the more recent period – positively to increases in oil price shocks (expectations of world economic recovery), negatively to an appreciation of the USD against major currencies (trade falls and thus GDP growth, reducing company earnings), and also negatively to real interest rates (according to the present value model). These findings are very robust to the use of an alternative stock market index and are much larger in the more recent period. The combination of interest rates and price pressures in goods markets provides most of the driving forces, helped by commodities and foreign exchange from external markets. We conjecture that flight to quality (toward

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U.S. assets) and improved liquidity (fueled by QE programs and the very low range for the overnight rate between 0 and 0.25%) have played important roles. In addition to sample correlations and VARs, the vine copula approach provides information on both the degree and the structure of dependence. Multivariate t-copula has been used repeatedly for modeling financial return data. It has been shown (e.g., Mashal & Zeevi, 2002) that the fit of multivariate t-copula is generally superior to that of other multivariate copulas for financial data. However, multivariate t-copula has only one parameter for modeling tail dependence, independent of dimension. In order to provide more flexible dependence modeling, Aas et al. (2009) propose paircopula method which is also known vine copula. They demonstrate that D-vine copula with t-copula for each pair is better than multivariate t-copula in modeling a data set consisting of the Norwegian stock index (TOTX), the MSCI world stock index, the Norwegian bond index (BRIX) and the SSBWG hedged bond index. After all, multivariate t-copula is only a special case of D-vine copula. Using the selecting procedure of Dißmann et al. (2013) and the goodnessof-fit test proposed by Schepsmeier (2013), we find that the most appropriate dependence structure for our data is a C-vine copula with various pair-copulas. Some of the pair-copulas in the C-vine copula incorporate tail dependences which are possibly asymmetric. As mentioned by Reboredo (2011) when investigating crude oil benchmark prices (WTI, Brent, Dubai, and Maya) using weekly data from January 1997 to June 2010, tail dependence “is a measure of the propensity of crude oil markets to go up or down together”. Examining contagion effects between daily return data from BRICs (Brazil, Russia, India, and China) and the U.S., Aloui, Ben Aïssa, and Nguyen (2011) show stronger dependence for commodity price dependent markets than for finished-goods export oriented markets. Applying a similar reasoning to our paper, tail dependence measures the propensity of real interest rates and oil prices to move together, with an important auxiliary channel being the exchange rates. An extension of this work, although necessarily relying on lower frequency data, is the role of fiscal policy (e.g., fiscal deficits) on asset returns (stocks and corporate and Treasury bond yields), documented by Jansen, Li, Wang, and Yang (2008) for the U.S. under monthly data. As mentioned in Section 1, vine copula can be applied in portfolio optimization in mainly two directions. Although there are some applications existing in the literature in these two directions, more applications can be explored and more interesting findings can be expected. We leave these topics for further research. References Aas, K., Czado, C., Frigessi, A., & Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44, 182–198. Abbara, O. Z. M. (2014). Assessing stock market dependence and contagion. Quantitative Finance, 14, 1627–1641. Akram, Q. F. (2009). Commodity prices, interest rates and the dollar. Energy Economics, 31(6), 838–851. Allen, D. E., McAleer, M., & Singh, A. K. (2014). Risk measurement and risk modelling using applications of vine copulas. Working Paper. Aloui, R., Ben Aïssa, M. S., & Nguyen, D. K. (2011). Global financial crisis, extreme interdependencies, and contagion effects: The role of economic structure? Journal of Banking and Finance, 35(1), 130–141. Arora, V., & Tanner, M. (2013). Do oil prices respond to real interest rates? Energy Economics, 36, 546–555. Arreola Hernandez, J. (2014). Are oil and gas stocks from the Australian market riskier than coal and uranium stocks? Dependence risk analysis and portfolio optimization. Energy Economics, 45, 528–536. Beare, B. K., & Seo, J. (2015). Vine copula specifications for stationary multivariate Markov chains. Journal of Time Series Analysis, 36, 228–246. Beckman, J., & Czudaj, R. (2013). Oil prices and effective dollar exchange rates. International Review of Economics and Finance, 27, 621–636. Bedford, T., & Cooke, R. M. (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence, 32(1), 245–268.

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