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5.2.2.1 Stationarity and Unit root tests

The estimation method of the standard regression model, Ordinary Least Square (OLS) method assumes that the mean and variances of the variables tested are constant over time. A variable with changing mean and variance over time is non- stationary or unit root variable. The inclusion of a non-stationary or unit root variable in estimating the regression equations with OLS method provides incorrect inferences. A stationary series has a constant mean, while non-stationary series has no constant mean. The impact of the random shock on non-stationary series tends to be permanent, and thus the series follows a random walk. A stationary data is required to derive a valid result from the estimation. Using non-stationary data in time series analysis leads to a “spurious” regression. That is a situation where an estimated regression will show a significant relationship while there is no any economic relationship between the variables (Glynn et al. 2007).

Hence, in time series data analysis, the concept of stationarity is very vital. Before undertaking any time series econometrics analysis, it is necessary to check the stationarity properties of the series. Augmented Dickey-Fuller test was commonly used to verify stationarity.

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5.2.2.2 Structural Breaks

Time series data often contain structural breaks, either because of change in policy or a sudden shock to the economy. Hence, the tests for parameter instability and structural change in regression models used to be an essential part of applied econometric studies since the work of Chow (1960), which tested for breaks a priori known dates with an F-statistic. So many other tests for a structural break like Quandt (1960), Andrews (1993), Andrews and Ploberger (1994), Bai (1997), Bai and Perron (1998, 2003) among others were developed over the years.

The idea behind Chow (1960) test is to fit each sub-sample separately to observe if their difference is significant in the estimated equations. If the difference is significant, it means there is a structural change in the relationship. Quandt (1960) amended the Chow framework to ease the condition that the candidate break date has to be known. Quandt (1960) consider the F-statistic that has the largest value over all possible break dates. More so, Andrews (1993) and Andrews and Ploberger (1994) derived the limiting distribution of the Quandt and related test statistics. Furthermore, Bai (1997), Bai and Perron (1998, 2003) built on the Quandt-Andrews context and presented a test which allows for multiple unknown breakpoints.

In this study, the combination of Chow, Quandt-Andrews, Bai-Perron tests are used to check for the existence of a structural break in each of the series used and the cointegrated VAR estimation.

5.2.2.3 Cointegration

Most economic time series are often non-stationary and I(1) series. A regression containing the levels of the non-stationary series produce misleading results which

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spuriously show a significant relationship between unrelated series. However, Engle and Granger (1987) pointed out that a linear combination of two or more non- stationary I(1) series can be stationary I(0), where such non-stationary series are cointegrated. It means there is the long-run equilibrium relationship between the variables. The concept of cointegration is relevant because differencing a variable to make it stationary only gives the short run dynamics. However, the study also examines the long-run relationships. With CVAR models cointegration test is essential given that the VECM will only be estimated when the variables are cointegrated.

5.2.2.4 VAR Lag Length Selection Criteria

Before estimating a VAR, the maximum lag length has to be decided to generate the white noise error terms. However, this is now done automatically by some econometric packages. The optimal lag length is determined using various information criteria. The commonly used information criteria are the Akaike (1974) information criterion (AIC), Schwarz’s (1978) Bayesian information criterion (SBIC) and the Hannan-Quinn information criterion (HQIC). Usually, the lag length suggested by most of the criteria is included the in the VAR system.

5.2.2.5 VAR Diagnostic tests

Some diagnostic tests were carried out after VAR models estimation. These diagnostic tests are essential to ensure that the obtained result from estimating the VARs are valid for policy analysis. The basic post-estimation test carried out on the residuals of VAR include LM test to check for serial correlation, the ARCH-LM test

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for heteroskedasticity in the VAR, Jarque-Bera test to check the normality and test for the stability of the VAR.

Stability Analysis

The VAR stability test is essential to ensure the result is valid for policy analysis. The stability test determines if the roots of the characteristic polynomial lie inside the unit circle. When all roots lie inside the unit circle, it indicates that the VAR is stable, and we can use it for policy analysis.

5.2.2.6 Impulse Response Function

Macroeconomic models are used for policy analysis just as are used for forecasting. When used for policy analysis, it is vital to know the sensitivity of the economy or any part of it to exogenous shocks. When researchers and policy makers use a model defining a relevant part of an economy, the often used tool for examining the impact of shocks is impulse response functions (IRF). The IRFs are also essential tools to model builders in analysing dynamic properties of models. It is also critical in the case of nonlinear models where it is not possible in most cases to work out the properties analytically Terasvirta et al. (2010, p.364).

IRF traces the impact of a variable on others in the system. According to Pesaran and Shin (1998), Impulse response system measures the time profile of shock’s effect at a given point in time on the (expected) future values of variables in a dynamical system. Therefore, for every variable from each equation, a unit shock to the error is evaluated to define the impact on the VAR system over time. For this study, the IRF will be used to examine the sign, size, and persistence of shocks from the consumer prices, import prices, and exchange rate.

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There are two methods often used to estimate impulse responses. They are the generalised impulse response function (GIRF) and the Cholesky decomposition. The Cholesky decomposition imposes a recursive causal structure from the top variables to the bottom variables but not the other way around. However, Cholesky decomposition is criticised for its sensitivity to the ordering of the variable in the model. It also means omitting an important variable would lead to distortions in the IRF.

Pesaran and Shin (1998) introduced the GIRF to avoid the problem of ordering dependence of the Cholesky decomposition IRF. The generalised impulse response does not require orthogonalization of innovations and is invariant to the reordering of the variables in the VAR (Pesaran and Shin, 1998).

5.2.2.7 Variance Decomposition

Variance decomposition measures the contribution of each type of shock to the forecast error variance. More specifically, it highlights the proportion of the movements in the dependent variables that are a result of their shocks, versus shocks from the other variables (Stock and Watson, 2001). Variance decomposition gives information about the relative importance of each random innovation to the variables in the VAR. In our case, variance decomposition shows the importance of shocks to the import and consumer prices themselves versus shocks from the exchange rate and other variables in the system.

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