A. The GVAR model framework
In following two sections, we discuss the econometric framework of the GVAR model, which was first proposed by Pesaran, Schuermann and Weiner (2004, hence PSW), and further developed by Dees, diMauro, Pesaran and Smith (2007, henceforth DdPS). The GVAR framework is well suited to examining spillovers because it allows us to model country-specific dynamics, while also allowing for cointegration among variables.
Assume an N number of countries i = 0, 1… N to be included in the model. For each country, a t number of domestic variables t = 1, 2 … t such as GDP, inflation, equity prices, etc… are collected into a xit : ki × 1 vector of domestic variables. Accordingly, an
With wij,j = 0,1,…,N a set of weights such that
Then for each country, a VARX*(2, 2) structure will be constructed, where:
With the ui,t cross-sectionally weakly correlated such that
We therefore can write the error correction form of the VARX*(2,2) specification as such:
βi is then partitioned as βi = (β’ix, β’ix*)’ conformable to zit, in order to write the ri error correction terms as
The foreign variables
Next, reduced rank regression is used in order to obtain the number of cointegrating relations ri, the speed of adjustment coefficients αi, and the cointegrating vectors βi for each country’s VARX*. The rank orders are obtained by Johansen’s trace statistic. Thus, each VARX* is estimated, allowing for cointegrating within the domestic variables xit, and between the domestic and foreign variables xit and
Once βi is estimated, the remaining parameters of each country’s VARX* can be obtained by OLS, using the following equation:
With ECMi,t–1 as the model correction terms according to the ri cointegrating relations of the ith country model.
The lag order for domestic variables (pi) and foreign variables (qi) that are included in each country’s VARX* model is selected using the Akaike Information Criterion (AIC), subject to a maximum lag order of pi = 2 that we chose.
Once each country-specific VARX* model is estimated, we can solve the GVAR model for the world as a whole: when the GVAR is solved for the world as a whole, all the variables become endogenous to the system, i.e. the GVAR model is expressed in terms of a k × 1 global variable vector,
By calling on
We then use the weights Wij, which we obtained from the trade or financial flows, in order to create the link matrix Wi. Using the link matrix Wi, we obtain the identity:
Where xt = (x’0t,x’1t,…,x’Nt)’ is the k × 1 vector of all endogenous variables of the system, and Wi is the
Finally, the individual models are stacked to yield the model for all the variables in the global model xt given by:
with p = max(max pi, max qi)
Premultiplying Equation 8 by
Equation (9) can be solved recursively and impulse response and variance decomposition analysis can then be performed.
B. GVAR estimation
We use the GVAR toolbox by Vanessa Smith and Alesandro Galesi15 in order to estimate our model. The order of integration for foreign and domestic variables is obtained by testing for unit root. All variables are tested with the Augmented Dickey-Fuller test as well as the Weight Symmetric ADF test on levels, first and second differences. Results are available upon request. We find that for the majority of the variables, the hypothesis of unit root cannot be rejected. All variables are tested for weak exogeneity, and the majority of foreign variables are weakly exogenous. Choosing to exclude non-exogenous variables from the VARX* specification has no statistically significant impact on the results. The trend coefficients are restricted to lie in the cointegrating space, and the intercepts are left unrestricted. (This is case IV in the GVAR toolbox).
In addition, our model satisfies the additional requirements indicated by PSW (2004):
The GVAR is stable: the eigenvalues of the F matrix in (9) lie on or inside the unit circle.
The weights are relatively small: PSW states that the weights must be small such that
as N → ∞. Most of our weights satisfy this condition.
The idiosyncratic shocks are weakly correlated. We can check for weak correlation by calculating the average pair-wise cross-section correlation between residuals and variables. The VECMX* residuals show low correlation among all variables.
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We would like to thank Elif C. Arbatli, Dennis Botman, Odd Per Brekk, Qianying Chen, Luc Everaert, Gee Hong Joong Shik Kang, Purva Khera, Juan Martinez, Colin McKenzie, Naoko Miyake, Yoshino Naoyuki, Masao Ogaki, Jack Ree, Niklas Westelius, Sun Yan and the participants in an IMF JISPA seminar for their helpful comments on this working paper. We would also like to thank Alessandro Galesi and Vanessa Smith for their technical help with the GVAR Matlab toolbox.
The consumer price index includes the temporary impact of the consumption tax increase.
Our sample includes Norway, Sweden, Switzerland, and the United Kingdom, and the Euro region, formed by pooling the following 8 countries: Austria, Belgium, Finland, France, Germany, Italy, the Netherlands, and Spain.
We also proxy the portfolio investment flows as a financial channel. However, results are not reported as this data is unavailable for many emerging Asian economies.
Galesi and Sgherri (2009) examine the propagation of shocks in Europe through a negative shock to the U.S. equity prices or GDP growth rate.
For example, the Chinese Renminbi to the Japanese Yen exchange rate is expressed as JPY/CNY.
Further details about the data are included in the data appendix.
We also tried the alternative ordering described in Dees di Mauro, Smith and Pesaran (2007) by placing the policy variable as the last in the ordering block, and the results are in line with our first ordering. We also tried placing equity prices as the second variable in the ordering block.
Placing the United States as the first country bloc will still yield very similar results.
All variables are in natural logarithm, and are first adjusted to inflation.
The decreasing impulse response function of Japan’s exchange rate reflects depreciation since we are using the nominal effective exchange rate for Japan.
Norway’s depreciation is presumably due to the Norwegian Kroner being a petroleum currency.
We also ran different scenarios as robustness checks, such as keeping the monetary base constant from 2103M1, 2013M2, and keeping equity prices constant in 2013M2 as well. Then, we also tested to see if results are homogeneous with the monetary base and equity prices kept constant simultaneously for 2013M1 and 2013M2. Results are strongly robust.
While Abenomics’ QQE was effectively launched in April 2013, expectations started rising in the first quarter of the year due to speculation.
The GVAR toolbox is available for download from wwwcfap.jbs.cam.ac.uk/research/gvartoolbox/index.html.