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Do house prices impact consumption and interest rate in South Africa? Evidence from a time-varying vector autoregressive model.

1. Introduction

The permanent income hypothesis asserts that house price inflation increases the expected lifetime wealth of homeowners and hence their desired consumption. This is known as the wealth effect. The collateral effect, on the other hand, postulates that fluctuations in house prices relax homeowners' financial constraints, which may in turn affect their actual consumption. Numerous papers indicate a strong positive link between the housing market and consumption in the United States (Green, 1997; Belsky and Prakken, 2004; Carrol, 2004; Iacoviello, 2005, 2011; Case et al., 2005; Leamer, 2007; Kishore, 2007; Jarocinski and Smets, 2008; Sousa, 2008; Vargas-Silva, 2008; Ghent, 2009; Mian and Sufi, 2009; Pavlidis et al., 2009; Ghent and Owyang, 2010; Iacoviello and Neri, 2010, Shirvani et al., 2012), the United Kingdom (Aoki et al., 2002; Campbell and Coco, 2007; Muellbauer and Murphy, 2008; Elbourne, 2008; Attanasio et al., 2011 ) and for other individual countries, such as Australia (Dvornak and Kohler, 2007), China (Chen et al., 2009; Koivu, 2012), Czech Republic (Sec and Zemcik, 2007), Hong Kong (Cheng and Fung, 2008; Gan, 2010), Italy (Paiella, 2004; Guiso et al., 2005; Bassanetti and Zollino, 2010; Bulligan, 2010), Portugal (de Castro, 2007; Farinha, 2008), Singapore (Edelstein and Lum, 2004; Phang, 2004), Spain (Aspachs-Bracons and Rabanal, 2011), Sweden (Chen, 2006) and Turkey (Akin, 2011). A few comprehensive international studies are also available, where a number of countries are studied concurrently (Boone et al., 2001; Bertaut; 2002; Bayoumi and Edison, 2003; Byrne and Davis, 2003; Barrel and Davis, 2004; Catte et al., 2004; Ludwig and Slok, 2004; Case et al., 2005; Aron and Muellbauer, 2006; Fung and Cheng, 2007; Goodhart and Hofman, 2008; Slacalek, 2009; Aron et al., 2010; Musso et al., 2011; Andre et al., 2011; Ciarlone, 2011; Peltonen et al., 2012; Sonje et al., 2012). (1)

As far as South Africa, our country of interest in this paper, is concerned, to the best of our knowledge, there exist four studies (2) analyzing the relationship between consumption and real house prices. They are Aron and Muell-bauer (2006), Das et al., (2011), Ncube and Ndou (2011) and Simo-Kengne et al. (forthcoming a). (3) Aron and Muellbauer (2006) indicated that much of the empirical literature assessing the wealth effect of house prices on consumption is marred by poor controls for the common drivers of both house prices and consumption. Given this, the authors suggested an empirical model for the United Kingdom and South Africa grounded in theory, and with more complete controls than generally used. The estimates suggested that in South Africa, unlike the UK, the marginal propensity to spend for housing wealth or collateral is slightly larger than for illiquid financial assets, though the difference is not statistically significant. Das et al., (2011), first tested for house price bubbles in the South African housing market, based on the unit root test developed by Phillips et al. (2011). Next, the authors estimated an error correction model to investigate the existence of spillover effects from the housing sector onto consumption. Results indicated significant spillovers, though there was no evidence of the effect being higher during the bubble period. In an attempt to understand the indirect channels through which monetary policy influences real variables by focusing on transmission to consumption using a structural VAR (SVAR), Ncube and Ndou (2011) provided evidence of significant spillovers from real house prices onto consumption. Finally, Simo-Kengne et al. (forthcoming a) provided an empirical analysis of the role of house prices in determining the dynamic behavior of consumption in South Africa using a panel vector autoregression approach applied to provincial level panel data covering the period of 1996 to 2010. With the shocks being identified using the standard recursive identification scheme, the study found that the response of consumption to house price shocks is positive, but short-lived. Thus, overall, irrespective of the methodology used, these three studies provided evidence of the existence of significant spillovers from the housing market onto consumption in South Africa, both at regional- and national-levels. These results are not surprising, given that in South Africa, housing, in general, accounts for 29.40 percent of household assets and 21.68 percent of total wealth (Das et al., 2011). Although these numbers are not as large compared to the U.S. economy, which happens to be 37.78% and 47.92% respectively (Iacoviello, 2011), they most certainly cannot be ignored.

Note that, price developments in the housing market affect asset value, which influences economic activity through the wealth channel. The wealth channel implies that a rise in asset prices increases the collateral value of the homeowner and subsequently their levels of credit access. Increased credit leads to money growth, higher consumption and ultimately inflationary pressure. Thus, in addition to studying the response of private consumption to a house price shock, the question of the response of monetary authorities to developments in house prices is also highly important and seems to have gained prominence among academics, especially in the wake of the recent financial crisis. Further, given the fact that the South African Reserve Bank (SARB) has moved to an official inflation-targeting framework since the first quarter of 2000, (4) there is clearly added value in analyzing this question for the country specifically. It seems logical for central banks to react to house price shocks insofar as they affect economic activity and inflation. But some economists advocate a more active role for monetary policy in preventing the development of bubbles that can be costly in terms of future output and financial stability (e.g. Roubini, 2006). Others argue that monetary policy is not the appropriate instrument to deal with asset bubbles (e.g. Posen, 2006). In some countries, central banks have occasionally referred to house prices as one of the parameters influencing monetary policy decisions (e.g. Australia, Sweden, United Kingdom). As central banks generally examine a wide set of economic variables to inform their policy decisions, it is difficult in practice to determine whether house prices play a role in interest rate setting. A number of recent studies (Castro, 2011; Naraidoo and Ndahiriwe, forthcoming and Naraidoo and Raputsoane, 2010 amongst others) have developed financial conditions indices (FCI), which include house prices amongst other financial variables, and have analyzed the importance of the FCI using linear and non-linear Taylor (1993)-type rules in the euro area, the UK, the US and South Africa. These studies tend to show that barring the US Federal Reserve, central banks have systematically reacted to the FCI, more so during the current financial crisis. The Ncube and Ndou (2011) study, discussed above, indicated a significant delayed response of monetary policy following a house price shock in South Africa.

Darracq Paries and Notarpietro (2008) and Finocchiaro and von Heideken (2009) analyze whether house prices play a role in the interest-setting behavior of central bankers using DSGE models, explicitly accounting for a housing sector, in the US and euro area, and Japan, the UK and the US, respectively. Their results suggest that trying to address the endogeneity problem in stand-alone monetary policy reaction functions augmented with house prices using General Method of Moments (GMM) methods produces biased and dispersed estimates. Thus, there are concerns using singleequation Taylor (1993)-type models. Furthermore, the studies using a FCI, which is a composite of four or five asset-related variables, does not specifically indicate the role of house prices in the monetary policy reaction functions. The studies using DSGE models tend to reach similar conclusions to those based on a FCI regarding the non-responsiveness of the Federal Reserve to house price movements. Some evidence of simultaneous interest rate response to house price shocks in Sweden and the UK have been provided by Bjornland and Jacobsen (2010) based on SVAR models, with monetary policy shocks being identified based on a combination of both short- and long-run restrictions. Prior to Bjornland and Jacobsen (2010), Elbourne (2008) too drew similar conclusions based on a SVAR for the UK. Musso et al. (2011) obtain similar results for the aggregate euro area, but indicated the lack of immediate interest rate response to house price shocks in the US. Prior to these studies, Demary (2010) also provided some evidence on the effect of a house price shock on interest rate, besides output, for ten OECD countries based on a SVAR model with shocks identified using the Choleski scheme.

Against this background, the objective of this paper is to not only analyze whether real house price movements have significant spillover effects on consumption decisions in South Africa, but also, whether house price shocks result in a simultaneous response in the monetary policy instrument, or whether the response is a delayed one following inflationary pressures due to an increase in aggregate demand resulting, in particular, from the wealth effect of a positive shock in real house prices. In addition, unlike the existing literature, which essentially relies on constant parameter models, we use a time-varying parameter vector autoregressive (TVP-VAR) model with stochastic volatility. TVP-VARs are quite common in the analysis of macroeconomic issues and allow us to capture the time-varying nature of the under-lying structure in the economy in a flexible and robust manner (Nakajima, 2011). Therefore, this paper attempts to analyze how consumer spending and monetary policy have reacted over time in South Africa to the evolution of house price shocks. To the best of our knowledge, this is the first attempt in the literature, using South Africa as a case study, to analyze the timevarying spillover effect of house price shocks on consumption and interest setting behavior, with the time-varying framework allowing us to not only identify the general relationship between the variables of interest, but more importantly, enables us to view how these relationships change depending on the underlying macroeconomic structure of the economy.

The decision to use South Africa as our country of investigation emanates from our familiarity with major structural changes and shifts in monetary policy regimes in the economy over the period of the analysis, and their possible effects on the variables under consideration in the TVP-VAR model. The remainder of the paper is organized as follows: Section 2 discusses the methodology of the TVP-VAR technique. Section 3 lays out the data used. Section 4 presents the results of a house price shock on consumption and the interest setting behavior. Finally, section 5 concludes.

2. Methodology

A vector autoregression (VAR), proposed by Sims (1980), has become a popular technique used in econometric analysis and is adaptable to a vast array of economic settings (Baltagi, 2011). In this study, a TVP-VAR model with stochastic volatility is used. The TVP-VAR is common in the analysis of macroeconomic issues and allows us to capture the time-varying nature of the underlying structure in the economy in a flexible and robust manner (Nakajima, 2011). The parameters in the VAR specification are assumed to follow a first order random walk process, thereby incorporating both temporary and permanent changes to the parameters. The inclusion of stochastic volatility is an important aspect in this TVP-VAR model. In many situations, a data-generating process of economic variables seems to have drifting coefficients and shocks of stochastic volatility. In that case, the application of a time-varying parameter model but with constant volatility may result in biased estimations of the time-varying coefficients, since a possible variation of the volatility in disturbances is ignored. The TVP-VAR model with stochastic volatility avoids this misspecification. Although stochastic volatility makes the estimation difficult due to the intractability of the likelihood function, the model can be estimated using Markov Chain Monte Carlo (MCMC) methods in the context of a Bayesian inference.

Following Nakajima (2011), this paper estimates a time-varying parameterVAR model with stochastic volatility of the form:

[y.sub.t] = [c.sub.t] +[B.sub.1t][y.sub.t-1] + ... + [][y.sub.t-s] + [e.sub.t], [e.sub.t] ~ N(0, [[ohm].sub.t]), (1)

for t = s + 1, ..., n, where [y.sub.t] is a ( k x 1) vector of observed variables, [B.sub.1t], ..., [] are (k x k) matrices of time-varying coefficients, and [[ohm].sub.t] is a (k x k) time-varying covariance matrix. A recursive identification scheme is assumed by the decomposition of [[ohm].sub.t] =[A.sub.t.sup.-1][[SIGMA].sub.t] [[SIGMA].sub.t][A'.sub.t.sup.-1]], where [A.sub.t] is a lower-triangle matrix with diagonal elements equal to one, and [[SIGMA].sub.t] = diang([[sigma].sub.1t], ..., [[sigma].sub.kt]). Let us define [[beta].sub.t] as the stacked row vector of [B.sub.1t],..., []; [a.sub.t] is the stacked row vector of the free lower-triangular elements of [A.sub.t]; and [h.sub.t] =([h.sub.1t],. ...,[h.sub.kt]) where [h.sub.jt] = long [[sigma].sub.jt.sup.2]. The time-varying parameters are assumed to follow a random walk process:


for t = s + 1, ..., n, with [e.sub.t] = [A.sub.t.sup.-1][[SIGMA].sub.t][[epsilon].subt] where [[SIGMA].sub.a] and [[SIGMA].sub.h] are diangonal, [[beta].sub.s+1 ~ N([[mu].sub.[beta]o], [[SIGMA].sub.[beta]o]), [a.sub.s+1] ~ N([[mu]], [[SIGMA]]), and [h.sub.s+1] ~ N([[mu].sub.ho], [[SIGMA].sub.ho]). (5)

A Bayesian inference is used to estimate the TVP-VAR models via MCMC methods. The goal of MCMC methods is to assess the joint posterior distributions of the parameters of interest under certain prior probability densities that are set in advance. We assume the following priors, as in Nakajima (2011): [[SIGMA].sub.[beta]] ~ IW(25, 0.01I), ([[SIGMA].sub.[alpha]]).sub.i.sup.-2] ~ G(4, 0.02), ([[SIGMA].sub. h]).sub.i.sup.-2] ~ G(4, 0.02), where ([[SIGMA].sub.[alpha]]).sub.i.sup.-2] and ([[SIGMA].sub. h]).sub.i.sup.-2] are the i-th diagonal elements in [[SIGMA].sub.[alpha]] and [[SIGMA].sub.h] respectively. IW and G denotes the inverse Wishart and the gamma distributions respectively. For the initial set of the time-varying parameter, flat priors are set such that: [[mu].sub.[beta]o] = [[mu]] = [[mu].sub.ho] = 0 and [[SIGMA].sub.[beta]o] = [[SIGMA]] = [[SIGMA].sub.ho] = 10 x I.

3. Data

The data sample covers the quarterly period of 1966:1 until 2011:2. A three-variable TVP-VAR model is estimated, capturing the time-varying nature of the macroeconomic dynamics in the South African economy between real consumption, nominal interest rate and real house prices. Seasonally adjusted real personal consumption expenditure data is obtained from the official website ( of the SARB, while the three-month Treasury bill rate data is derived from the International Financial Statistics of the International Monetary Fund, as is the seasonally-adjusted Consumer Price Index (CPI) data, used to convert nominal house prices into its real counterpart. Amalgamated Bank of South Africa (ABSA), one of the major private banks in South Africa, provides the seasonally adjusted house price index. Based on all the standard unit root tests, namely, Augmented Dickey-Fuller (1981) (ADF), Phillips-Perron (1988) (PP), Dickey-Fuller test with generalized least squares detrending (DF-GLS), the Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) (1992) test; the Elliot, Rothenberg, and Stock (ERS) (1996) point optimal test, the Ng-Perron (2001) modified versions of the PP (NP-MZt) test and the ERS point optimal (NP-MPT) test, real consumption expenditure and real house prices were found to be non-stationary, so the variables were converted to their corresponding growth rates, and denoted as DC and DRHP. The nominal interest rate was found to be stationary at the 10 percent level of significance using ADF, DF-GLS, ERS, NP-MZt and NP-MPT tests, and hence, was used in levels, and denoted as TBILL. (6) The stable (7) TVP-VAR is estimated based on two lags, as was unanimously suggested by all the popular lag-length tests, namely, the sequential modified LR test statistic, the Akaike information criterion, the Schwarz information criterion, applied to a constant parameter VAR. Accounting for stationarity and lags, our effective sample period start from 1966:4.

4. Results

To compute the posterior estimates, we draw M = 10,000 samples after the initial 1,000 samples are discarded. Table 1 presents the estimates for the posterior means, standard deviations, 95 percent credible intervals, (8) the convergence diagnostics (CD) of Geweke (1992) and the inefficiency factors of selected parameters of the TVP-VAR, computed using the MCMC sample. (9) Based on the CD statistics, the null hypothesis of the convergence to the posterior distribution in the estimated result is not rejected for the parameters at the 5 percent level of significance. In addition, the efficiency factors are quite low. Finally, the 95 percent confidence intervals include the estimated posterior mean for each of the parameters estimated. Therefore, the results show that the MCMC algorithm produces posterior draws efficiently. Figure A, in the appendix, presents the estimation results of the TVP-VAR model with stochastic volatility.
Table 1 Estimation results of selected parameters in the TVP-VAR model

Parameter                            Mean     Std       95%     CD
                                             Dev.   Interval

[([[summation].sub.[beta]]).sub.1]  0.0665  0.0146  [0.0439,  0.590

[([[summation].sub.[beta]]).sub.2]  0.0483  0.0076  [0.0353,  0.589

[([[summation].sub.a]).sub.1]       0.1094  0.0511  [0.0480,  0.319

[([[summation].sub.a]).sub.2]       0.0819  0.0338  [0.0432,  0.370

[([[summation].sub.h]).sub.1]       0.1787  0.0683  [0.0830,  0.539

[([[summation].sub.h]).sub.2]       0.4640  0.1013  [0.2901,  0.350

Parameter                           Inefficiency

[([[summation].sub.[beta]]).sub.1]         28.52

[([[summation].sub.[beta]]).sub.2]          9.92

[([[summation].sub.a]).sub.1]              34.34

[([[summation].sub.a]).sub.2]              54.52

[([[summation].sub.h]).sub.1]              35.55

[([[summation].sub.h]).sub.2]              39.39

Note: The estimates of
[[summation].sub.[beta]] and [[summation].sub.a] are multiplied by 100.


Figure 1 presents the data of the three variables (DC, TBILL and DRHP) in the top panel, with the bottom panel plotting the corresponding posterior estimates of stochastic volatility. The time-series plots consist of the posterior draws on each date. Stochastic volatility of growth in consumption spikes around 1976, followed by general downward trend. Since 1990 it has remained stable and low. The low stochastic volatility towards the end of the sample period may reflect more certainty in consumption behavior derived from a stable economic and political environment. The Treasury bill rate exhibits two major spikes in stochastic volatility during the 1979 oil crisis and in 2000 when inflation targeting was implemented as the new mandate of monetary policy. Stochastic volatility in the growth of real house prices tends to show a general downward trend over the sample period, with cyclical variations around this trend. The significant posterior estimates of the stochastic volatility present in the variables of interest, justifies the use of a TVP-VAR model with stochastic volatility to avoid biased estimation.


Impulse responses are used as a tool to capture the macroeconomic dynamics in the estimated VAR system. For a standard constant parameter VAR model, the impulse responses are drawn for each set of two variables, whereas for a TVP-VAR model, the impulse responses can be drawn in an additional dimension, as the responses are computed at all points in time using the time-varying parameters. There are several ways to simulate the impulse responses based on the parameter estimates of the TVP-VAR model. Following Nakajima (2011), we compute the impulse responses by fixing an initial shock size equal to the time-series average of stochastic volatility over the sample period, and using the simultaneous relations at each point in time, for considering the comparability over time. In the VAR, the variables are ordered in an attempt to try and identify the housing demand shock using a recursive or Choleski identification scheme, as obtained based on the lower-triangular matrix At. We order the variables as follows: DC, TBILL and DRHP following Musso et al. (2011) and Andre et al. (2011). In this regard, note that the equation for the real house price can be interpreted as a housing demand function, which, in turn, relates the real house price to consumption and interest rate. As in Jarocinski and Smets (2008), Iacoviello and Neri (2010) and Musso et al., (2011), a non-monetary housing demand shock is such that an increase in real house prices leads to a rise in consumption through time without being associated with a fall in the monetary policy instrument, so that we can distinguish the shock from an expansionary monetary policy shock. Furthermore, it is assumed that consumption does not react simultaneously to this shock, so that the shock cannot be dubbed a positive technology shock, including of the "positive news" type shock. To compute the recursive innovation of the variable, the estimated time-varying coefficients are used from the current date to future periods. Around the end of the sample period, the coefficients are set constant in future periods for convenience. Although a time series of impulse responses for selected horizons or impulse responses for selected periods are often exhibited in the literature, one could draw a three-dimensional plot for the time-varying impulse responses.

Figure 2 shows the impulse responses for the growth rate of consumption and the nominal interest rate following a housing demand shock obtained from the constant-parameter VAR model, while Figure 3 plots the corresponding time-varying responses for the TVP-VAR model. The latter responses are drawn in a time series manner by showing the size of the impulses for one-quarter, four-quarter, eight-quarter, and twelve-quarter horizons over time. The time-varying nature of the macroeconomic dynamics between the variables is shown in the impulse responses, and, to some extent, the shape of the impulse response in the constant VAR model is associated with the average level of the response in the TVP-VAR model. A one standarddeviation positive shock to DRHP causes the growth rate of real house prices to stay significantly positive for six-quarters, before becoming negative and insignificant from the ninth-quarter. The positive shock to DRHP cases DC to increase significantly for five quarters, before becoming negative and insignificant from the seventh-quarter ahead. The nominal interest rate response is a delayed one, with the TBILL only starting to increase from the second quarter, with the effect becoming significant only after a year. The interest rate effect is quite persistent in that it stays on being significantly positive for the remainder of the horizon. (10)



Figure 3 illustrates the (TVP)-VAR model's impulse response trajectories at different horizons of one-quarter, four-quarters, eight-quarters and twelvequarters at each point of the sample. It illustrates what the relationship is like between the variables following a house price shock over time. In general, a housing price shock is observed to have a positive relationship with consumption over time. However, a stronger positive relationship is experienced in the short-term as shown by the one-quarter ahead trajectory, which is consistently positive over the entire sample. For the longer steps (eight and twelve), the effect is small and becomes negative after around ten quarters from the beginning of the sample, which, in turn is understandable given similar behavior of the real house price growth rates for these horizons during this period. The effect on consumption in the medium to long-run horizons started to show a positive trend from around 1990, five years or so after the financial market liberalization in 1985, following the de Kock Commission report. Note the effect of house price shocks on house price itself post-1985 has been consistently positive over all the horizons. A further round of strong positive effects of real house price growth rates on consumption was observed starting in early 2000 when the housing market in South Africa was booming. Understandably, towards the end of the sample as financial crisis set into the country, with real house prices becoming negative and the trajectories of the real house price impulse responses becoming flatter, the effect on consumption started to fade away. Clearly then, over the period of the study, there exists quite a degree of variability in the behavior of consumption following a house price shock. Treasury bill rates generally exhibit a positive, but delayed, response to a house price shock. The one-period-ahead trajectory is virtually zero for the entire sample, and in fact, no relationship exists between the two variables at the shortest horizon from the early 1990s. The responses tend to get stronger as the horizons increase. The monetary policy response is found to decline post the financial liberalization until 2000, (11) when the SARB moved to an official inflation-targeting regime. Post 2000 and until the financial crisis, the SARB does seem to have responded positively to increases in house prices. The monetary policy response to house price movements is found to have been weaker during the financial crisis, when the real house price growth rates became negative. This seems to suggest that the SARB was not keen on raising interest rate during the financial crisis following positive house price shocks to allow the housing demand to grow. (12) In short, we find that the wealth channel is prevalent in South Africa, and has been especially dominant after the financial liberalization, and also monetary authorities have been consistently responding to house price shocks.


5. Conclusion

This paper uses a three variable (growth rate of real consumption, nominal three-months Treasury bill rate and real house price growth rate) TVP-VAR model with stochastic volatility to analyze the impact of a house price shock on consumption levels and monetary policy for South Africa over the quarterly period of 1966:4-2011:2. We find that the impact of house price shocks on consumption is found to have increased post financial liberalization in 1985 - a result which makes sense, since the liberalization of the domestic markets led to less stringent conditions for obtaining mortgage credits. Monetary policy response to house price shocks became weaker post financial liberalization, until the SARB moved to the official inflationtargeting regime. The effect of house prices on both consumption and interest rate was understandably weak during the financial crisis. On average, as can be seen from the constant-parameter VAR model, real house price shocks had a significant impact on consumption growth in South Africa that lasts for more than a year. Furthermore, a delayed but significant and persistent monetary policy response is observed following a positive real house price shock.


We would like to thank Jouchi Nakajima for many helpful comments. The usual disclaimer applies.


(1.) Another strand of the literature focuses on international spillovers (see for example Otrok and Terrones, 2005; de Bandt et al., 2010 and Vansteenkiste and Hiebert, 2011).

(2.) South Africa was part of the panel of developed and developing economies considered by Fung and Cheng (2007) and also one of the fourteen emerging countries in the panel of Peltonen et al. (2012).

(3.) Related to these studies, Simo-Kengne et al. (forthcoming b) empirically examined the effect of house price changes on economic growth across provinces in South Africa. The economic impact of house prices was estimated using a panel data set that covered all nine provinces in South Africa from 1996 to 2010. The authors found that when heterogeneity, endogeneity and spatial dependence are controlled for, house price changes exhibit a significant effect on regional economic growth in South Africa.

(4.) In the February of 2000, the Minister of Finance announced that inflation targeting would be the sole objective of the SARB. Currently, the Reserve Bank's main monetary policy objective is to maintain CPI inflation between the target-band of three to six percent, using discretionary changes in the repo rate as its main policy instrument.

(5.) For a comprehensive analysis of the TVP-VAR methodology and the estimation algorithm, refer to Nakajima (2011).

(6.) These results are available upon request from the authors.

(7.) The constant parameter VAR is found to be stable as all roots were found to lie within the unit circle.

(8.) Bayesian inference uses "credible intervals" as opposed to "confidence intervals" used in the frequentist approach to highlight parameter uncertainty.

(9.) Geweke (1992) suggests the comparison between the first n0 draws and the last n1 draws, dropping out the middle draws, to check for convergence in the Markov chain. The CD statistics are computed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [x.sub.(i)] being the i-th draw, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the standard error of [[bar.x].sub.j] respectively for j = 0, 1. If the sequence of the MCMC sampling is stationary, it converges to a standar normal distribution. We set m0=1, n0=1000, m1=5001, and n1=5000. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is computed using a Prazen window with bandwidth (Bm) = 500. The inefficiency factor is defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[rho].sub.s] is the sample autocorrelation at lag s, which is computed to measure how well the MCMC chain mixes.

(10.) The impulse response functions for a shock to DC and TBILL were found to be in line with standard economic theory. A positive consumption shock (aggregate demand shock) led to a rise in the interest rate and real house prices. A contractionary monetary policy shock reduced consumption and real house prices. These results are available upon request from the authors. Interestingly, unlike in the literature, we did not observe a price puzzle, where by real house prices are found to rise temporarily following a contractionary monetary policy shock. The interested reader is referred to Gupta et al. (2012a) for further details.

(11.) The response of monetary policy post financial liberalization declined until inflation targeting was adopted, since the percentage change of house prices for one percentage point increase in house prices increased during this period - a phenomenon commonly observed in the literature (Gupta et al., 2012b) and also seen from the impulse response of house prices following a contractionary monetary policy.

(12.) As with the constant parameter VAR, the impulse response functions for a shock to DC and TBILL were, in general, found to be in line with standard economic theory over the entire sample. The details of these results are available upon request from the authors.


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Author:Peretti, Vittorio; Gupta, Rangan; Inglesi-Lotz, Roula
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Date:Dec 1, 2012
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