Data was first acquired from the OECD website where a minimum of 100 data points was collected. 112 data points were collected for Canada's quarterly GDP data from 1980 quarter 1 to 2007 quarter 4. From this, the data was split into two periods; a test period and an estimation period. The last thirty observations was used as a test period and treated as ex-post; the other eighty-two values were used as the estimation period. An in-sample forecast was then made using the Naive trend and the Naive rate of change, Decomposed time series model, Brown's double exponential smoothing model, Holt's two-parameter method and a non-seasonal ARIMA model.
To gain the lowest Mean Squared Error (MSE) of each in-sample forecast, the parameters of each model was altered until this was achieved. Each model was used to create a thirty one-step short-term forecast and twenty-seven four-step long-term forecasts on the ex-post data. The in-sample models were tested by the performance of the MSE. The Theil-U statistic was also used to support the performance of the MSE. The best model that produced the best short-term and long-term ex-post forecast was the ARIMA (0 1 2), as it was the model to have both lowest MSE along with a minimum p-value of 10%.
As a result, this model should be used for ex-ante forecast. For Time Series Analysis, models are forecasted to find the future value in the series. This is achieved by looking at past data points in a series to make future observations and for leading indicators, which use current and up to date data that are available to them from other data series. Time Series Analysis are an important tool for firms that work on the stock market as economic agents use it to forecast future values to help them with their financial strategies.
This is frequently used in business, economics and finance. An in-sample forecast is where forecasts are made on the selected period of data available, and from this, a model is formulated. In-sample forecasts are used to decide the parameters of the model and in this report; it will be seen as the 'fits'. Ex-post forecast is where the data is available to make a forecast period but not to make a model. The ex-post forecast can be used to create n-step forecasts, for example, it can be used to make a one-step short-term forecast and twenty-seven long-term forecasts.
These forecasts are used to test the accuracy of the models as an instant comparison between forecast and actual data is possible. Ex-ante forecast is used to test the future period when there is no data available. With ex-ante forecast, you can only make one forecast for n steps at a time. These types of forecasts are leading indicators to forecast the future. The aim for a business or government is either to make these forecasts as accurate as possible for future investment plans, to set the interest rate, or to predict what the inflation would be. When we forecast the data, it requires us to use time series analysis.
The data is collected from the OECD website of constant prices. Once acquired, the data is split up into two periods; the estimation period and the test period. The test period will be of the last thirty observations, and will be treated as ex-post. The ex-post forecasts will be of thirty one-step short-term forecast and twenty-seven four-step long-term forecasts, of different models and it will be examined against the ex-post to see which is more accurate. From the estimation period, we can use this for the ex-post forecast to find the parameters of the model.
The Brown and Holt's model have weighted averages to their model therefore we will need to find the optimal value by using different weighting constants. Different weighting constants give us different forecast and the most accurate forecast will be used as the ex-post. The forecast models will produce short-term and long-term forecasts. The short-term forecast will be thirty-one one-step forecasts, which will emulate a monthly forecast. The twenty-seven four-step forecast will be long-term and will show a yearly forecast.
After each forecasted model, the MSE will be calculated to assess the performance of each model. The data, which was collected, had to have enough observations to make a reliable forecast. For this reason a minimum of 100 observations were collected. The last thirty observations was the test period and treated as ex-post, which was used for the in-sample forecast. The other seventy observations would be used as the estimation period. The data that was collected had to be of constant prices of GDP and did not include inflation, otherwise the data would fluctuate and it would make it difficult to forecast the data.
The data that was a collected needed to be accurate because there are a variety of different sources where the information could be collected and not give us a reliable data. The data also had to be from a single source because collecting data from numerous sources would have made the forecast unreliable, and lead to statistical problems. To satisfy these criteria the data was collected from the OECD website. 1 Figure 1: Canada's GDP Whole Realisation From Figure 1, the data starts from 1980 Q1 and has 123 observations to end at 2010 Q3.
The data is to be split into two periods, an estimation period and a test period. The estimation period is of the whole realisation whereas the test period of going to be of the last 30 observations. From the graph, we can see that there is a recession that occurs and therefore will be difficult to forecast the data, shown by the green dashed line. As a result, the forecast will be from 1980 Q1 up until 2007 Q4. The ex-post data is shown by the red dashed line, which is going to be the test period. The test period starts from 2000 Q3 to 2007 Q4.