During the last decade significant attention has been paid to find the best way to characterize or model the dynamic properties of economic time-series. As outlined by Culver and Pappel (1997) the dominant topic in time-series econometrics and the issue in empirical macroeconomics is the distinction between unit root and stationary processes. Stationary time-series can be characterized as series whose statistical properties such as mean, variance, and autocorrelation are all constant over time (Fuller, 2009).
Therefore the series is believed to be stationary (at least in weak sense) if: - E( ) is constant
- Var ( is constant
- Cor ( = thus it changes as changes
However, since not all series can be found to be stationary, most statistical forecasting methods are based on the assumption that the time-series can be return to stationarity by using specific mathematical operations (Asteriou and Hall, 2011). In stationary series the shocks are temporary and their effects will be eliminated as the series revert to their long-run mean values. In contrast to stationary series, non-stationary time-series necessarily contain permanent components. Thus, a serious and very common problem of forecasting macroeconomic time-series is that they are often trended or affected by persistent innovations to the process, in other words, they are non-stationary (Brooks, 2008). Moreover, the use of non-stationary data can lead to spurious regressions. If standard regression techniques are applied to non-stationary data then as a result, the standard Ordinary Last Squared (OLS) regression procedures could easily lead to incorrect conclusions. The end result could be a regression with significant coefficient estimates and a high but valueless R2 (Brockwell and Davis, 2009). Such a model would be termed as a spurious regression. In other words, since this model requires time-series data when the dependent variable is the interest rate followed by explanatory variables such as inflation rate and commodity prices, which are in most of the cases non-stationary, the problem of spurious regression might be highly possible. A
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consequence of overlooking the problem of spurious regression could be arriving at wrong conclusions and having misleading results of estimation. This gives a rise to one of the main reasons for taking the logarithm of data before subjecting it to formal econometric analysis. Taking the logarithm of a series, which exhibits an average growth rate, will turn the time- series into the series following a linear trend and is integrated (Mahadeva and Robinson, 2004):
(4.2)
(4.3)
Therefore the lagged dependent variable has a unit coefficient, which in each period increases by an absolute amount equal to .
As discussed previously, due to the characteristics of time-series, the non-stationarity is often presented; therefore the first step should include the identification of whether the time-series is stationary or non-stationary. Non-stationarity can be characterised in two ways, by the random walk model with drift:
(4.4)
Random walk is non-stationary because even its mean is constant, it grows linearly in time therefore it is difficult to predict variance of since Var( )= . The second model to characterize non-stationarity is the deterministic trend process:
(4.5)
The deterministic trend process is non-stationary because it changes over time and (white noise) has no memory of the past (Chatfield, 2004).
An effective and generally used method for testing non-stationarity is a unit root test. If the time-series is stationary, then it is acceptable to continue in estimating regression as shown in Figure 4.1 however, if not, the non-stationarity of time-series cannot be ignored. To get round the problem of non-stationary series, it is common to test whether series are stationary in levels or differences (Clements and Hendry, 1998). The unit root test is one of the tests of stationarity. The pioneering work on testing for a unit root in time-series was done by Dickey and Fuller (1979), where the basic objective of the test is to examine the null hypothesis that (thus the series contains a unit root and is non-stationary) in an autoregressive (AR)
87 equation of the form:
(4.6)
against the one-sided alternative . Therefore the hypotheses of interest are H0: Series contains a unit root versus H1: series is stationary.
A more convenient version of the test can be obtained by subtracting from both sides of the equation:
(4.7)
Where .
There are three types of equations in the Dickey-Fuller test (or Augumented D-F) also known as the -test awhich can be conducted allowing for an intercept, intercept and deterministic trend or neither:
Without intercept or trend: , E( )=0 the assumption is that the series fluctuate around a flat line which is equal to zero.
Intercept: ,E( )= the assumption is that series fluctuate around a flat line which is not equal to zero.
Intercept and trend: , E( the assumption is that the series fluctuate around a flat line which is not equal to zero and has a trend.
However, the D-F test does not take into account possible autocorrelation in the error process . Present autocorrelation in can also cause inefficiency of the OLS estimates (Cochrane, 1991). There are two main drawbacks of the D-F test. First, the test is only valid if is white noise. In particular, will be autocorrelated if there is autocorrelation in the dependent variable of the regression . Due to this limitation the power of the D-F test is, according to Cook (2004), only 25 per cent. In other words in the D-F test the possibility for misinterpretation of unit root results is 75 per cent.
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A more comprehensive theory of unit root and non-stationarity has been developed by Phillips and Perron (1988) who incorporated an automatic correction to the D-F procedure to allow for autocorrelated residuals. However, the most important criticism of the original Dickey-Fuller and Phillips-Perron-type tests is that their power is low if the process is stationary but with a root close to the non-stationary boundary. The tests are poor at deciding especially with small sample sizes, as the null hypothesis can be either rejected or not rejected. A failure to reject the null hypothesis could occur either because the null hypothesis was correct or due to insufficient information in the sample to enable rejection (Zhang, 2008). Therefore Dickey and Fuller (1981) developed the Augmented Dickey-Fuller (ADF) test which approximates the autocorrelation based on the following regression model:
(4.7)
This model is augmented and the solution to serial correlation is to include k –number of lags for . Number of lags is crucial as including too many lags can increase the error in the forecasts or estimation results. However, including too few could leave out relevant information. Information criterion procedures can help to sort out the problem with an appropriate number of lags. Three commonly used information criterions are: Schwarz's Bayesian information criterion (SBIC), the Akaike' information criterion (AIC), and the Hannan and Quinn information criterion (HQIC) (Ivanov and Kilian, 2001).
If there is an unknown trend status, or the information that a trend exists is used, then SIC and AIC always perform better than the two simulated hypothesis testing strategies. In case of monthly data, the Akaike information criterion (AIC) is preferred since it performs better in smaller samples with higher frequency (Hacker, 2010).
As already noted, not all time-series data can be found to be stationary. Since there are two characterisations of non-stationarity, the random walk with drift and the trend-stationary process (Equation 4.4 and Equation 4.5), they also require different treatments to induce stationarity. To achieve stationarity in the random walk with drift it is necessary to subtract
from both sides:
(4.8)
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Equation 4.8 shows the induction of stationarity by differencing. Equation 4.10 represents deterministic non-stationarity, therefore stationarity is achieved by detrending (Clements and Hendry, 1995). If it is assumed that all time-series data are stationary or stationarity can be achieved by detrending, the natural procedure is to continue by estimating the Vector Autoregressive model (VAR) or its structural form (SVAR). The next part is therefore dedicated to the introduction of autoregression modelling.