### Vectorautoregressive- VAR Models and Cointegration Analysis

Implementation ardl long run analysis and pss bound test in eviews 9 seems to be If you want to test for an existence of a longrun relationship, you can use the . In multivariate causality tests, the testing of longrun causality between two variables is more problematic, as it the null hypothesis of non-stationarity concludes “cointegration relationship” does exist. . significance at 1%, ** at % levels with respect to Pesaran and Pesaran () critical values. Extra credit Quiz pdl-inc.info Testing Finance Growth Nexus: An Auto Regressive Distributed Lag (ARDL) We used F-test to test the existence of long run relationship; this test has an asymptotic real deposit rate (RDR) we apply ARDL cointegration technique introduced by Pesaran et al. These were in the form of quiz, treasure hunts and so on.

The long run real equity price equation can be derived from a log-linear approximation of the rst order Euler equation in consumption-based asset pricing models, or can be obtained more directly from present value relations.

Also, because of the Fisher relationship and the term structure conditions, the UIP can be considered equally in terms of long-term interest rates. The latter is de ned in terms of the US dollar exchange rates, whilst the former is measured in terms of the bilateral exchange rates. However, they do not permit the speci cation and testing of PPP.

This is because in a multi country set up as noted above the PPP is best formulated in terms of e ective exchange rates ee it rather than the US dollar rate, e it. Tests of the PPP hypothesis do not depend on the choice of the reference country. It is, therefore, established that if A is non-singular and PPP holds for all real e ective exchange rates then it must also hold in terms of the US, and more generally for any country pairs.

It is also worth noting that nonsingularity of A means that trade weights are such that no country or group of countries is isolated from the rest of the world economies. This result is particularly pertinent when N is relatively large and a full system approach to the analysis of cointegration along the lines suggested by Johansen 99 might not be possible.

By focussing on possible Similarly, the long run relations y it y it Iand S it S it cointegration of y it and yit for each i we are also able to shed light on the possibility of pair-wise cointegration Pesaran, Using z it, 3. For example, in the case of the above illustration x t is a vector, whilst there are 9 endogenous variables in the global model.

To deal with the problem of exchange rate modelling in a closed system rst using 3. Note that we are now solving for the US price level and not the US in ation rate, although it is in ation that is being solved for in the case of the other countries. Persistence Pro les The persistence pro les PP refer to the time pro les of the e ects of system or variable-speci c shocks on the cointegrating relations in the GVAR model, whilst the impulse responses refer to the time pro le of the e ects of variable-speci c shocks or identi ed shocks such as monetary policy or technology shocks identi ed using a suitable economic theory on all the variables in the model.

The impulse responses of shocks to speci c variables are known as the generalized 12 impulse response functions GIRF. In the context of the GVAR the cointegrating relations are given in terms of the country-speci c variables, namely iz it, whilst the variables in the GVAR are given by x t, and appropriate mappings between z it and x t should be used. See DdPS for further details. In fact there is evidence that on average the shocks across countries are positively correlated.

For the derivation of the generalized forecast error variance decomposition see Appendix A. GFEVD can also be performed for the errors of the global model, " t. After a brief review of the GVAR model used, we present the results of the tests for the long run restrictions imposed, before looking at the persistence pro les of the implied GVAR model.

Based on this model, we analyse the transmission of shocks to oil and equity prices as well as monetary policy shocks in the global economy through impulse response analysis and forecast error variance decomposition. This allows us to empirically evaluate the e ects of imposing the theory-based long run restrictions on the short run as well as the long run properties of the model.

### Bounds Testing Approaches to the Analysis of Long-run Relationships

Finally, we show how the results change when the alternative de nition of the exchange rate viz a viz the US dollar is used. All tables and gures can be found at the end of the paper. As noted earlier the endogenous variables included in the country speci c models are the logarithm of real output y it ; the quarterly rate of in ation, it, the real e ective exchange rate, re it ; the short-term interest rate, S it ; and if relevant real equity prices, q it, and the long-term interest rate, L it.

The trade shares used to construct the country-speci c foreign variables the "starred" variables are given in the trade-share matrix provided in a Supplement to DdPS available on request. Table 2 presents the trade shares for the eleven focus economies ten countries plus the euro areawith the "Rest" category showing the trade shares for the remaining countries.

With the exception of the US model, all individual models include the country-speci c foreign variables, yit ; it ; q it ; S it ; L it and oil prices p o t. The country-speci c foreign variables are obtained from the aggregation of data on the foreign economies using as weights the trade shares in Table 2. Because the set of weights for each country re ects its speci c geographical trade composition, foreign variables vary across countries. We use xed trade weights based on the average trade ows computed over the three years It is clearly possible to use di erent types of weights for aggregation of di erent types of variables.

The problem is one of data availability and empirical feasibility. We have addressed this issue in DdPS partly by considering time-varying trade weights. Also in the case of equity and bond prices that tend to move very closely across di erent economies it is unlikely that using other weights could matter much.

Subject to appropriate testing, the country-speci c foreign variables are treated as weakly exogenous when estimating the individual country models.

The concept of weak exogeneity in the context of the GVAR is discussed in DdPS and relates to the standard assumption in the small-open-economy macroeconomic literature. In this case the star variables are said to be long run forcing for the domestic variables, and implies that the error correction terms of the individual country VECMs do not enter in the marginal model of the foreign variables.

We provide in DdPS a formal test of this assumption for the country-speci c foreign variables the "starred" variables and the oil prices. Recall that the speci cation of the US model di ers from that of the other countries in that oil prices are included as an endogenous variable, while only re US;t ; y US;t ; and US;t are included included in the US model as weakly exogenous.

The endogeneity of oil prices re ects the large size of the US economy. The omission of qus;t, S US;t and L US;t from the vector of US-speci c foreign nancial variables re ects the results of tests showing that these variables are not weakly exogenous with respect to the US domestic nancial variables, in turn re ecting the importance of the US nancial markets within the global nancial system. Finally, the issue of parameter instability is also dealt with in DdPS, where we conduct a number of structural stability tests along the lines of Stock and Watson and nd that although there is evidence of structural instability, this is mainly con ned to error variances and do not seem to adversely a ect the coe cient estimates.

In view of changing error variances we use robust standard errors when investigating the impact e ects of the foreign variables, and base our analysis of impulse responses on the bootstrap means and con dence bounds rather than the point estimates.

The number of cointegration relationships is derived from cointegration tests see DdPS. The tests yield a number of 3 cointegration vectors for most of the eleven focus countries, except China only one vector and the US 2 cointegrating vectors. In the case of the UK and Norway, while the tests indicate that 4 cointegrating vectors could not be rejected borderline5 17 we decided to impose only 3 cointegrating relations based on the persistence pro les and impulse responses.

The former allow us to check whether a restriction corresponding to a long run relationship is valid by converging to zero and whether it produces reasonable speed of convergence. Once the number of cointegrating relationships is determined, we proceed to incorporate the long-run structural relationships, suggested by economic theory as outlined in Section 2 in our otherwise unrestricted country-speci c models.

The over-identifying restrictions are imposed simultaneously on the countries, while the remaining 5 individual country VECM models are estimated subject to just-identifying restrictions.

We also experimented by imposing over-identifying restrictions on each of the countries separately, imposing just-identifying restrictions on the remaining countries.

The results obtained were very similar. Table 3 reports the long-run restrictions that correspond to each country, for the case where the in ation coe cient in the Fisher equations is restricted to unity in all the focus countries.

The choice of the possible restrictions is arrived at based on a satisfactory performance of the GVAR model in terms of stability eigenvaluespersistence pro les and impulse response functions.

According to the value of the t-statistic on the in ation coe cient, we then determine the country models for which the Fisher equation can be left unrestricted. It is worth noting that the value of the in ation coe - cient can in this case be interpreted as the importance of the in ation term in the Central Bank feedback rule.

In accordance with the Taylor principle, the coe cient on in ation should be greater than one if the Central Bank wants to ensure that the real interest rates move in the right direction to stabilize output. In the euro area, Canada and Australia, the coe cient on in ation is not signi cantly di erent from one.

For the remaining countries, China, Sweden, Switzerland, Norway and New Zealand, the in ation coe cient was estimated to be less than one. This is a di cult result to interpret and requires further investigation, at least in the case of the latter four economies. However, as recently argued by Nelson 25 the low estimate of the in ation coe cient in the case of some of these countries, New Zealand in particular, could be explained by the extensive use of price and wage controls during 98 s and early 99 s.

Building on the initial results reported in Tables 3 and 4, the nal set of over-identi ed long-run restrictions for the focus countries are summarized in Table 5.

## Pesaran testing for the existence of a long-run relationship quiz

The critical values reported are computed by bootstrapping from the solution of the GVAR model see Appendix A for the computational details. The results in Table 5 show that only in the case of 9 We also considered the long run real equity price equation, 2. In all other cases the long run relations are not rejected by the data. Furthermore, all the long run relations have well behaved the persistence pro les see Figure indicating that the e ects of shocks on the long run relations are transitory and die out eventually.

It is interesting that this property holds even in the case of the long run relations for the three countries with LR statistics above their bootstrapped critical values; thus providing some support for the validity of the long run relations entertained even for these economies.

Overall, the test results support the term premium condition i. L S s I in nine out of the ten focus countries where the condition is relevant there are no long run rates in China. Strict Fisher hypothesis is supported in the case of euro area, Canada and Australia, with the less strict version of the hypothesis holding for all the remaining economies.

But, strict PPP can only be detected for three countries the UK, Australia, Norway and a weaker form with relative productivity di erences cannot be rejected in the case of Switzerland. Hence, among the eleven focus countries, we reject both absolute and relative PPP only in the case of the US and China.

For the US, this result can be explained by the role of the US dollar as a reserve currency. As proposed by Juselius and MacDonald 23the peculiar role of the US dollar has facilitated relatively cheap nancing of the large US current account de cits explaining why an adequate adjustment toward PPP between the USA and the rest of world has not taken place.

In the case of China, as the country has remained in transition towards the market economy over the period, it is therefore not surprising that such "market failures" can be found.

### Pesaran testing for the existence of a long-run relationship quiz – zyknia

Regarding the contemporaneous e ects of the foreign variables on their domestic counterparts, as in DdPS we continue to nd only weak linkages across the short-term interest rates, s and s ; with Sweden no longer constituting an exception. The contemporaneous elasticity of real equity prices remains signi cant and slightly above one in most cases as in the unrestricted case, while we also continue to observe signi cant linkages across the long-term rates with the exception of New Zealand.

In terms of real output the elasticity of UK real output with respect to yuk;t is now more in line with the rest of the countries increasing to. In contrast, the real output elasticity of Australia decreases from. Finally, in ation elasticities show the greatest Alternative restrictions and speci cations chosen did not appear to alter this result. In particular, in ation elasticity in Japan with respect to the foreign in ation is now signi cant dropping from.

The in ation elasticity in the euro area remains signi cant and at. Thus, in most cases it appears that the imposition of the Fisher equation tends to reduce in ation elasticities, showing a higher degree of independence of domestic in ation from their foreign counterparts. Turning to the e ectiveness of the country speci c foreign variables in reducing the cross-section correlation of the variables, we deal with this as in DdPS, by computing average pair-wise cross-section correlations of the country speci c residuals over the estimation period.

What is worth noting, is that now with the use of the real e ective exchange rate we no longer observe high correlations for the exchange rate variable after conditioning on the foreign variables. The main diagonal terms of the B0 matrix the coefficients on the ith variable in the ith equation are scaled to 1. That is, the structural shocks are uncorrelated. A plot of the row i, column j element of as a function of lag l is called the non-orthogonalized impulse response function.

X can help in predicting Y.

Regression of X on Y has a big R2 2. Y can not help in predicting X. In the I 0 case the solution is: A stock market index and the price of its associated follow a random walk by time. Testing the hypothesis that there is a statistically significant connection between the futures price and the spot price could now be done by testing for a cointegrating vector. The usual procedure for testing hypotheses concerning the relationship between nonstationary variables was to run Ordinary Least Squares OLS regressions on data which had initially been differenced.

Although this method is correct in large samples, cointegration provides more powerful tools when the data sets are of limited length, as most economic time-series are.

The two main methods for testing for cointegration are: The Engle-Granger three-step method. Error Correction Model Granger Representation Theorem Determination of the dynamic relationship between cointegrated variables in terms of their stationary error terms. Regression with only stationary variables on both sides. Following the estimation the hypotheses H0: The Johansen test is used to test for the existence of cointegration and is based on the estimation of the ECM by the maximum likelihood, under various assumptions about the trend or intercepting parameters, and the number k of cointegrating vectors, and then conducting likelihood ratio tests.