The econometric model is simple. The structural form of the time-varying vector autoregression model with lag length p for a process ytis given by
A0yt= γ + A1yt−1+ A2yt−2+ · · · + Apyt−p+ ut (5.1) where we allow for all parameters, including intercept coefficients, autoregressional coefficients and the covariance matrix of stochas-tic terms to be contingent on the unobservable state variable st ∈ S
(to ease the burden of notation we suppress the state-dependent subscripts). The vector autoregression model is chosen because it imposes (relatively) fewer presumptions on the data structure, and it also conveniently parameterises the dynamic interactions within a system.4 The time-varying coefficients capture possible non-linearities or time variation in the lag structure of the model.
The stochastic volatility allows for possible heteroscedasticity of the stochastic terms.
The variables of interest yt= (y1,t, y2,t, . . . , ym,t)are an m×1 vector.
The stochastic intercept term γ = (γ1(st), γ2(st), . . . , γm(st)) cap-tures the difference in the intercept under different states. A0 is an m× m state-dependent matrix which measures the contemporane-ous relationship between variables, and the econometric identifica-tion of the model is obtained through restricidentifica-tions on A0. Ak is an m× m matrix with state-dependent elements a(ij)k (st), i, j= 1, . . . , m, k= 1, . . . , p. The stochastic error term utwill be explained below.
The corresponding reduced form of the above model can be obtained by pre-multiplying Equation 5.1 by A−10 , which yields
yt= d + Φ1yt−1+ Φ2yt−2+ · · · + Φpyt−p+ εt (5.2) where d= A−10 γ, Φk= A−10 Akand εt= A−10 ut, k= 1, 2, . . . , p. Φk is an m× m matrix with state-dependent elements φ(ij)k (st), i, j= 1, . . . , m, k = 1, . . . , p. We further define d(st) ≡ c + α(st), which will be explained below. The stochastic error-term vector εt can be further expressed as
εt = A0−1ut= Λ(st)H1/2vt(st)
where H is an m× m diagonal matrix with diagonal elements σj2, j= 1, . . . , m, Λ(st)is an m×m diagonal matrix with diagonal elements λj(st), j= 1, . . . , m
Λ(st)=
⎡
⎢⎢
⎢⎢
⎢⎣
λ1(st) 0 · · · 0 0 λ2(st) · · · 0 ... ... . .. ... 0 0 · · · λm(st)
⎤
⎥⎥
⎥⎥
⎥⎦
which captures the difference in the intensity of volatility, and vt(st) is a vector of standard normal distribution, vt(st) ∼ N(0, Σ(st)),
where the covariance matrix is given by
In CCL 2008, two cases are considered. One is a three-variate time-varying SVAR(p) model, ie, m = 3. The three variables of interest are yt = (FFR, SPR, RET), where FFR denotes the federal funds rate, SPR is the interest rate spread and RET denotes either REIT returns (REIT) or housing market returns (HRETs). Essentially, they study the asset returns “one at a time”. They also compare the case of the stock return based on the S&P 500 Index (SRET).
They then extend this to the case of the four-variate model, where yt= (FFR, SPR, GDP, RET); GDP denotes GDP growth. It turns out the results are pretty robust despite the introduction of the GDP growth variable. It is consistent with the notion that asset prices are forward looking and therefore information about (past) GDP growth has already been reflected in the asset price data. Hence, the intro-duction of the GDP variable will only marginally affect the SVAR of asset returns. For econometric identification, restrictions on the ele-ments of A0need to be imposed. Following the discussion in Leeper et al (1996) and Christiano et al (1998, 1999), A0 is specified to be a lower triangular matrix. In the three-variable case, it means that
A0=
and the four-variable case is similar. As shown in Equation 5.4, we have imposed a recursive restriction so that y1,t (FFR) affects y2,t (SPR), and both y1,tand y2,taffect y3,t(RET) contemporaneously, but not vice versa. On the other hand, it is still possible for RET to affect FFR and SPR, but with a time lag. Thus, the restriction may not be as stringent as it seems.
Two-state Markov process
Following the literature on Markov switching, and being limited by the sample size, we assume that there are only two states, ie, st∈ S = {1, 2}. The procedure for the identification of the regime of
the economy for a given period will be discussed below. The Markov-switching process relates the probability that regime j prevails in t to the prevailing regime i in t− 1, Pr(st = j | st−1 = i) = pij. The transition probability matrix is then given by
P=
p11 1− p11
1− p22 p22
The persistence can be measured by the duration 1/(1− pii), and hence the higher the value of pii, the higher the level of persistence.
Given that the economy can be in either state 1 or state 2, the term αj(st), j = 1, . . . , m, defined above, captures the difference in the intercept under different states. For convenience, we set αj(1)= 0 for st= 1; thus, αj(2) measures the difference in the intercept between state 2 and state 1. Furthermore, we set the diagonal element of Λ(st)at state 1 to be unity, ie, λj(1)= 1, so that if λj(2) > 1, then the intensity of volatility in state 2 is larger than that in state 1, and vice versa.
Since vt(st)is a vector of standard normal distribution and λj(1) is set to be 1, the variance of yj,t, j= 1, . . . , m, at state 1 is σj2, and the variance is λ2j(2)σj2.
Identification of regimes
Finally, we discuss the identification of regimes in this model. Since the state of the economy is unobservable, we identify the regime for given a time period by Hamilton’s (1989, 1994) smoothed probability approach, in which the probability of being state stat time t is given by π(st| ΩT), where ΩT= {y1, y2, . . . , yt, . . . , yT}. The idea is that we identify the state of the economy from an ex post point of view, and thus the full set of information is utilised. Notice that we only allow for two regimes, ie, st ∈ S = {1, 2}. Thus, if π(st = j | ΩT) >0. 5, then we identify the economy most likely to be in state j, j= 1, 2.
Stationarity of the Markov regime-switching model
The stationarity test of Markov regime-switching model is provided by Francq and Zakoian (2001). To illustrate the idea, take a three-variable, VAR(2) model as an example. Let
Γ (st)=
⎡
⎢⎢
⎣
Φ1(st) Φ2(st) 03 I3 03 03 03 03 03
⎤
⎥⎥
⎦
Figure 5.2 Smoothed probabilities for the SVAR(1) model of (FFR,
0 1980 1990 2000 1980 1990 2000
(a) (b)
(a) Regime 1; (b) regime 2.
Figure 5.3 Smoothed probabilities for the SVAR(1) model of (FFR, SPR, GDP, HRET) and Φ2(st)are the autoregression matrixes in Equation 5.2. We then define the following matrix and let ρ(Ξ) be the spectral radius of Ξ. Francq and Zakoian (2001) show that a sufficient condition for second-order stationarity of a Markov-switching VAR(2) model is ρ(Ξ) < 1.
Empirical results
This section highlights some of the empirical results of CCL 2008. Fig-ures 5.2–5.4 provide a visualisation of the identification of regimes
Figure 5.4 Smoothed probabilities for the SVAR(1) model of (FFR, SPR, GDP, SRET)
1.0 0.8 0.6 0.4 0.2 0
1980 1990 2000 (a)
1.0 0.8 0.6 0.4 0.2 0
1980 1990 2000 (b)
Table 5.3 AIC values for various three-variable VAR(p) models of the REIT system
VAR model State-contingent parameters P = 1
Single-regime model None 10.625
Two-regime model (A) c(st),Λ(st) 9.951 Two-regime model (B) c(st),Λ(st),vt(st) 9.928 Two-regime model (C) c(st),Λ(st),Φk(st) 9.945 Two-regime model (D) c(st),Λ(st),Φk(st),vt(st) 9.916
Note: The three variables are FFR, SPR and REIT.
Table 5.4 AIC values for various three-variable VAR(p) models of the HRET system
VAR model P = 1 P = 2
Single-regime model 6.120 6.084 Model A (two-regime model) 5.403 5.059 Model B (two-regime model) 5.402 4.972 Model C (two-regime model) 6.330 4.907 Model D (two-regime model) 5.308 4.781 The three variables are FFR, SPR and HRET.
in the four-variable case. Regime 1 is always the (relatively) “high-volatility regime”. It is clear that the identification of regimes is similar and yet they are not exactly identical. It is consistent with the notion that the asset returns tend to co-move and, at the same time, there are instances or periods in which some asset returns are
Figure 5.5 Impulse responses of REIT to innovations in FFR when the effect of SPR or GDP is shut off (FFR, SPR, GDP, REIT)
1.0
(a) Single regime; (b) regime 1; (c) regime 2. Solid line: full effect; dashed line:
SPR shut-off; dot-dashed line: GDP shut-off.
affected differently by the shocks. It should be noted that, given the current formulation, there is an asset-specific innovation and hence it is possible that the system of different asset returns would identify different periods. Tables 5.3 and 5.4 also confirm that the statistical model D, which allows all the parameters (including those in the variance–covariance matrix) to be regime dependent, performs best.
This confirms that regime switching is indeed an important feature of both the REIT and housing return during the sampling periods.
Figures 5.5–5.7 summarise a series of counter-factual analyses.
Notice that in this SVAR system the monetary policy (FFR) can affect the asset return in different ways. On top of the “direct effect” from FFR to the asset return, it can also influence the asset return through the terms spread or the GDP growth (the “indirect channel”). To dis-entangle the two effects, CCL 2008 first report the impulse response of the full effect, assuming that the system does not switch to an alternative regime. In other words, all reported impulse responses are “conditional”. They then shut down the effect from the term
Figure 5.6 Impulse responses of HRET to innovations in FFR when the effect of SPR or GDP is shut off (FFR, SPR, GDP, HRET)
0.1 0
−0.1
−0.2
−0.3
−0.4 5 10 15 20 25 30
(a) 0.1
0
−0.1
−0.2
−0.3
−0.4 5 10 15 20 25 30
0.1 0
−0.1
−0.2
−0.3
−0.4 5 10 15 20 25 30
(b)
(c)
(a) Single regime; (b) regime 1; (c) regime 2. Solid line: full effect; dashed line:
SPR shut-off; dot-dashed line: GDP shut-off.
spread to the asset return, and allow only a once-and-for-all inno-vation in the monetary policy variable (FFR). Hence, that second impulse response only records the direct effect and the indirect effect through GDP. The third case is when the effect from GDP to the asset return is shut down. Again, only a once-and-for-all innova-tion is allowed in the monetary policy variable (FFR). Hence, that third impulse response only records the direct effect and the indi-rect effect through the term spread. Figure 5.5 reports the case of REIT when a linear SVAR is estimated (single regime) and when a regime-switching SVAR is estimated. In the latter, CCL 2008 further distinguish the case when the system is under regime 1 from that under regime 2. The result is clear. When the term spread channel is shut down, the impulse response of REIT return “dies out” much more quickly under the single regime and under regime 2. In other words, the term spread acts like a “multiplier” in the case of REIT return.
Figure 5.6 repeats the exercise for the housing return. Interest-ingly, the case of “full effect” and that of “GDP effect shut off” are
Figure 5.7 Impulse responses of SRET to innovations in FFR when the effect of SPR or GDP is shut off (FFR, SPR, GDP, SRET)
1 0
−1
−2
−3
5 10 15 20 25 30
1 0
−1
−2
−3
5 10 15 20 25 30
1 0
−1
−2
−3
5 10 15 20 25 30
(a) (b)
(c)
(a) Single regime; (b) regime 1; (c) regime 2. Solid line: full effect; dashed line:
SPR shut-off; dot-dashed line: GDP shut-off.
remarkably similar, suggesting that the GDP plays a minor role in the propagation mechanism. On the other hand, under regime 1 (ie, the high-volatility regime), the impulse response is much more volatile when the term spread channel is shut down. In other words, the term spread acts like a “stabiliser” in the case of housing return.
Figure 5.7 repeats the exercise for the stock return (based on the S&P 500 Index). Interestingly, the case when the GDP growth is shut down and the case when the term spread is shut down provide very similar pictures to the case with the “full effect”, suggesting the very minor role of both for the monetary policy variable to affect the stock return. After all, the effect of the monetary policy on the stock return seems to be very small and short lived in all cases.
While some of the results in CCL 2008 are not discussed here due to limited space, there are still a few lessons we can learn from their exercises. First, the regime-switching nature seems to be very impor-tant in the data, especially for the asset return. Second, GDP does
not seem to play an important role in the propagation of the mone-tary policy to the asset market returns. Third, while the term spread seems to play an important role in the propagation of the monetary policy to the asset market returns, such a role seems to be asset spe-cific and regime dependent. One possible explanation is that, when the central bank cuts the (short-term) interest rate, the market par-ticipants anticipate a possible change in the long-term interest rate and hence the long-term interest rate tends to increase. The house-hold sector seems to be very aware of the long-term interest rate increase, while the corporate sector seems to focus on the benefit of the short-term interest rate, possibly reflecting the difference in source of financing. Thus, the policy makers may wish to be mindful of such a difference in their policy operation.