The variables of interest are oil imports to Germany, and temperature in Germany. The latter is used as a leading indicator for the former, to improve on the forecast obtained by the univariate model. Both variables are collected over a time range from January 1985 until and including December 1997, whereas the last year is not used for constructing the optimal forecast, obtained by fitting a model through the data until the end of 1996. This will enable us to forecast the year 1997 using our model, and then comparing it to the actual data.
Assuming no large one time shock, meaning that it is not captured by seasonality or cyclical behaviour in the data, occurs in this year, a graphical comparison of our forecast and the whole data range (including 1997) will tell us something about the preciseness of our model. Both variables are of monthly frequency. On the one hand this will leave us with enough observations (144) to have a reliable forecast; on the other hand, due to the nature of the data it makes intuitive sense to use monthly data at the project at hand.
The data series are not seasonally adjusted. Univariate model Data Inspection First we will smooth the series by transforming the data on oil demand into their logarithmic form. The log transformation allows the model to be less vulnerable to outliers in the data, and thus enables for a more precise forecasting model. Next the data series must be checked for trend and seasonality. Figure 1. 1 shows the time series plot for the log transformation of oil imports in Germany from 1985M01 until 1996M12.
pic] Before fitting a trend and seasonal dummies to the model, we must include an intercept, because we are obviously not starting at 0.
The time path of oil imports in Germany is clearly trending upwards through time. This makes intuitive sense as population is growing, increasing the demand for oil. Even so it appears to be definite that we have a linear trend, the series will be regressed using a linear, a quadratic and exponential trend.
The H0 of the test is that the data has a unit root against the Ha, that the data has no unit root. Table 1. 1 shows the result of the augmented Dickey-Fuller test. The p-value of the test is less than 0. 01%, so we can clearly reject H0, so our data has no unit root. Following model has been fitted so far: Now, knowing that the data has no unit root, we can start fitting an ARMA model. Fitting an ARMA model We will try to make an even better fit to our model by using the moving average process and autoregressive process. The aim is to end up with a stationary model, characterized by no significant correlation between the dependent variable and its lags. White noise zero conditional mean constant conditional variance.
We therefore will first have a look at the correlogram of our model when only including an intercept, the trend and seasonality. Both the autocorrelation and partial-autocorrelation show damped oscillation. (intuition) The partial autocorrelation has only one peak outside the 95% confidence interval at the first lag.
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