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3.4.3.1 Unit Root Tests

There is the possibility that economic and financial time series may contain at least one unit root31, being non-stationary; although some series are stationary and some series possibly contain two unit roots. Under this circumstance, using time series data is likely to cause significant errors, such as spurious regression problem. Therefore, as a first step, the stationarity property of any time series used should be checked.

The Augmented Dickey-Fuller (ADF) unit root test and the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test are used. The ADF test has the null hypothesis of non- stationarity, whereas the KPSS test has the null hypothesis of stationarity. By testing for a unit root using the two methods with different assumptions, the robustness of the unit root tests can be strengthened. In considering Figure 3.6, it is assumed that only GDP has a deterministic time trend, the others have stochastic time trends. So, both a deterministic time trend and an intercept are included in the test equation for GDP, while only an intercept is included in the test equation for other variables, such as government spending and net taxes32. The optimal lag lengths are selected as zero using the Schwarz information criterion (SIC).

31

Technically speaking, ‘unit root’ refers to the root of the polynomial in the lag operator. ‘Unit root’ indicates that a given time series is not stationary.

32 _{The results of unit root tests are the same even if it is assumed that GDP, government spending and net taxes}

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Figure 3.6 GDP, Government Spending and Net Taxes

Note: All the variables are in log of real values, and per capita terms

Table 3.2 displays the results of unit root tests for both the level and the first difference of each of the three variables in the baseline SVAR model. As expected, all of three level variables have a unit root, and so follow a non-stationary process. In other words, we can say that three variables are integrated of order one, i.e. I(1). According to the results of the ADF test, the null hypothesis of non-stationarity for all three level variables cannot be rejected, even at the 10 per cent significance level. The KPSS test rejects the null hypothesis of stationarity for all three level variables, presenting the same information as the ADF test.33

After all the level variables have been differenced, no unit root will be found in the first differenced variables. For all the first differenced time series, the null hypothesis of non- stationarity is rejected even at the 1 per cent significance level.

33 _{We also performed unit root tests with various options – only intercept, intercept and trend, or neither. In each}

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Table 3.2 Unit Root Tests Results of Baseline VAR Variables

Test statistics Variables

Levels First difference

verdict

ADF KPSS ADF KPSS

GDP

Real -1.988 0.187** -6.251*** 0.142* I(1)

Log (Real) -0.648 0.183** -5.536*** 0.102 I(1)

Government Spending

Real -2.139 0.757*** -6.773*** 0.461* I(1)

Log (Real) 1.465 0.764*** -6.558*** 0.487** I(1)

Net Taxes

Real 0.842 0.732** -6.122*** 0.296 I(1)

Log (Real) -1.767 0.764*** -6.500*** 0.325 I(1)

Notes: 1. ***, ** and * denote the rejection of the null at the 1%, 5% and 10% significance levels.

2. In a KPSS test, the spectral estimation is Bartlett kernel and Bandwidth is selected by Newey-West’s method. 3. The optimal lag lengths are selected as zero using the SIC.

3.4.3.2 Cointegration Test

Engle and Granger (1987) suggested that a linear combination of a number of non- stationary series can be stationary. These non-stationary time series are said to be ‘cointegrated if there is a stable long-run relationship between them by moving together over time even though individually each variable is non-stationary’ (Gujarati, 2004). We can say that ‘a cointegrating relationship is a long-term equilibrium, because cointegrating variables are able to diverge from their relationship in the short run, but their ties would get back in the long run’ (Enders, 2010, p356).

In this context, if there is no cointegration, we can construct a model of stationary variables by taking the first difference of the non-stationary variables. On the other hand, if there is a cointegrating relationship between the variables, we can estimate a level SVAR model without taking any difference on the variables. We conduct cointegration tests in two ways: one is Engle-Granger’s residual test, and the other is Johansen’s cointegration test.

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Firstly, using the ADF test on the logarithm of the ratio of net taxes to government spending, i.e., ln (net taxes / government spending), it is possible to see if there is a cointegrating relationship between the two variables. The null and alternative hypotheses for the unit root test are as follows.

(3.2) ̂ ( ) vs. ̂ ( )

Thus, the null hypothesis assumes a unit root, while the alternative hypothesis assumes no unit root. Under the null hypothesis, therefore, a stationary linear combination of the non-stationary variables has not been found. If the null hypothesis is rejected, it can be said that a cointegrating relationship exists between variables.

Figure 3.7 shows the log of net taxes over government spending. According to the ADF test, the null hypothesis is rejected at the 1 per cent significance level. We can conclude that there is a cointegrating relationship between net taxes and government spending.

Figure 3.7 Gap between Government Spending and Net Taxes

Table 3.3 Unit Root Tests Results of Gap

Notes: 1. ***, ** and * denote the rejection of the null at the 1%, 5% and 10% significance levels, respectively. 2. Gap = log of real net taxes per capita – log of real government spending per capita

3. In the tests, gap series are tested with both time trend and intercept

-0. 2 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1975 1980 1985 1990 1995 2000 2005 2010

Gap=log(net taxes)-log(goverment spending)

Test statistics Variable Levels verdict ADF KPSS Gap -6.797*** 0.132* I(0)

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Secondly, the issue of cointegration can be examined within the Johansen VAR framework. Although the above Engle-Granger approach is very easy to use, one of its major drawbacks is that it can only estimate one conitegrating relationship between the variables. We have three variables in our model, so more than one linearly independent cointegrating relationship can be witnessed. In this context, it is much more suitable to check the issue of cointegrating ralationship using the Johansen VAR framework.

The baseline SVAR model uses a set of three variables, which are I(1) according to the unit root test above. Before executing Johansen’s cointegration test, the appropriate number of lags in the SVAR system should be specified. For this, considering the annual frequency of the data set, we presume that the appropriate number of lags is no more than four. The Akaike information criterion (AIC), Schwarz information criterion (SIC) and Hannan-Quinn information criterion (HQ) suggest a VAR(1) with a constant.

Table 3.4 VAR Lag Order Selection Criteria

Lag With Constant No Constant

AIC SIC HQ AIC SIC HQ

0 -8.913604 -8.780289 -8.867584 NA NA NA

1 -14.97668* -14.44341* -14.79259* -14.60811 -14.20817* -14.47005* 2 -14.79637 -13.86316 -14.47422 -14.65217* -13.85228 -14.37605 3 -14.83753 -13.50438 -14.37733 -14.57435 -13.37451 -14.16017 4 -14.61043 -12.87733 -14.01217 -14.41633 -12.81655 -13.86409

Note: * indicates lag order selected by the criterion

So, a VAR with 1 lag containing three variables could be set up as follows. (3.3) +

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In order to use the Johansen cointegration tests, we have to change the VAR above into a vector error-correction model (VECM) of the form,

(3.4)

where .

Basically the VECM has variables in the first differenced form on the LHS, and lagged level terms on the RHS. As the model recommended above has only one lag length, it has only lagged level terms on the RHS. is a long-run coefficient matrix. The cointegration test between the variables is calculated by considering the rank of the matrix via its eigenvalues. The rank of a matrix is equal to the number of its eigenvalues that are different from zero. If the variables are not cointegrated, the rank of will be similar to zero (Enders, 2010, pp 385-392).

In the Johansen approach, the trace test ( ) and maximum eigenvalue test ( ) are used to determine the number of cointegrating vectors (r).

The statistics for the two tests are as follows. (3.5) ( ) ( ̂ ) and

(3.6) ( ) ( ̂ )

where T is the number of observations, r is the number of cointegrating vectors under the null hypothesis and ̂ is the estimated value for the ith ordered eigenvalue from the long-run coefficient matrix (Enders, 2010, pp 391-392).

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‘ is a joint test where the null hypothesis is that the number of cointegrating vectors is less than or equal to r against an unspecified or general alternative that there are more than r’(Enders, 2010, p 391). The test statistic has a sequence of null alternative

hypotheses as follows.

vs

vs

vs

‘ conducts separate tests on each eigenvalue, and has as its null hypothesis that the number of cointegrating vectors is r against an alternative of r+1’ (Enders, 2010, p 391). Therefore, test statistic has null and alternative hypotheses as follows.

vs

vs

vs

We keep increasing the value of r until we no longer reject the null hypothesis. The results appear as in Table 3.5. In total, five models are made based on whether an intercept and a trend are included or not. The results across the five types of model and the two types of test are the same, having one cointegrating vector.

Table 3.5 Selected Number of Cointegrating Relations (5 per cent significance level)

Data Trend None None Linear Linear Quadratic

Test Type No Intercept No Trend Intercept No Trend Intercept No Trend Intercept Trend Intercept Trend Trace 1 1 1 1 1 Max-Eig 1 1 1 1 1

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The results of the cointegration test after selecting ‘intercept (no trend) in cointegrating equation and test VAR and 1 lag length’ are shown below in Table 3.6.

Examining the trace test, if we look at the first row below the headings, the statistic of 43.72 by far exceeds 29.80 - the critical value of the 5 per cent significance level, and we can reject the null hypothesis of no cointegrating vectors. In the next row, we cannot reject the null hypothesis of at most one cointegrating vector because the trace statistic 14.87 is smaller than 15.49 - the critical value of the 5 per cent significance level. In conclusion, we can say there is one cointegrating vector in our VAR model. The maximum eigenvalue test confirms this result too.

Table 3.6 Johansen Maximum Likelihood Cointegration Test

Trace Test( ) Number of

cointegrating vector Eigenvalue Trace Statistic

5% Critical

Value Prob.**

None * 0.541499 43.71886 29.79707 0.0007

At most 1 0.311664 14.86653 15.49471 0.0620

At most 2 0.027923 1.047852 3.841466 0.3060

Maximum Eigenvalue Test( ) Number of

cointegrating vector Eigenvalue

Max-Eigen Statistic 5% Critical Value Prob.** None * 0.541499 28.85233 21.13162 0.0034 At most 1 0.311664 13.81867 14.26460 0.0587 At most 2 0.027923 1.047852 3.841466 0.3060

Note: * denotes rejection of the hypothesis at the 5 per cent significance level **MacKinnon-Haug-Michelis (1999) p-values

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