Use To Forecast Future Developments In The Economy

Published Date: 02 Nov 2017

Where ln is the natural log, tot_demt is the demand for electricity in time period t for Japan, incomet-1 is the income in period t-1, real_pricet is the deflated nominal price in period t and D is the change in demand between t and t-1. In the data set used, demand for electricity is measured in GWh and real prices are measured using the CPI index. The income measure is real disposable income in 1960 prices.

The following equation is the long run equation for electricity demand as it explains the long run behaviour of demand. l measures the long-run speed of adjustment. The long run relationship between demand and income was also tested.

A time series is a series of values of a quantity obtained at regular intervals over a period of time (daily values, weekly values, monthly values, quarterly values, yearly values etc.). Time series analysis is used in many areas such as economic forecasting, stock market analysis, sales forecasting, are firstly seasonal variations which occurs when the time series exhibits regular fluctuations during the same month, every year or during the same quarter every year. The second component is trend variations; the trend can be either positive or negative depending on whether the time series exhibits an increasing long term pattern or a decreasing long term pattern. If a time series does not show an increasing or a decreasing pattern then the series is stationary in the mean. The third component is cyclical variations which are patterns showing an up and down movement around a given trend. The duration of a cycle depends on the type of business or industry being analysed. The final component is random variations, this component is unpredictable and do not fall under any of the above three classifications. Time series forecasting predicts long term trends and seasonal fluctuations. Smoothing techniques are used to reduce random fluctuations in times series data. Smoothing can help remove seasonality and make long term fluctuations in the series to be clearer.

The data can be declared to be time series using the tsset command in Stata:

Declare Time series in Stata

Autocorrelation

Seasonal patterns of time series can be estimated via correlograms. An autocorrelogram describes a correlation between values of a process at different time intervals. The values can range from +1 to -1 . An auto correlation of +1 shows a perfect positive correlation compared to an autocorrelation of -1 which shows a perfect negative correlation. A stochastic process is considered to be weak stationary if its mean, variance and auto covariances are constant over time. Weak stationarity can be tested by the correlogram of a time series which is the graph of the autocorrelation at various lags. A partial autocorrelation is a correlation between two variables when the effects of one or more related variables are removed.

There is a high autocorrelation function for demand industry but it gradually declines. This correlation is non stationary. The partial correlation function dies out after lag 1.

There is a high autocorrelation function but it gradually declines. This correlation is non-stationary. The partial correlation function tends towards 0 after lag 1.

There is a high autocorrelation function for real price industry but it gradually decreases. This correlation is non stationary. The partial correlation function becomes 0 after lag 1.

There is a high autocorrelation function for real price household but it gradually decreases. This correlation is non stationary. The partial correlation function becomes 0 after lag 1.

There is a high autocorrelation function for income but it gradually declines. This correlation is non stationary. The partial correlation function dies out after lag 1 but increases again at lag 21 to lag 24.

A unit root test is a statistical test for stationarity or non stationarity for a time series. A well known test for a unit root in a time series sample is the Augmented Dickey-Fuller test (ADF). This statistical test involves determining whether or not the test statistic obtained is more extreme than the critical value. If the test statistic is greater than the critical value, then this leads to a rejection of the null hypothesis in favour of the alternative hypothesis and the null hypothesis is not rejected if the test statistic is less than the critical value. An economic time series can either be trend stationary or difference stationary. A trend stationary time series has a deterministic trend whilst a difference stationary time series has a stochastic trend. The ADF tests can be applied to determine whether the time series is trend or difference stationary. This model is a random walk with drift.

Unit root test for Demand (Industry)

The null hypothesis is that this test has a unit root. As shown in the table above, the test statistics for industry demand for electricity is -3.032 and the p value is 0.0320. As the 1st critical value is greater than the test statistics, the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Demand (Households)

The null hypothesis is that this test has a unit root. The above table shows that the test statistics for households demand for electricity is 0.326 and the p value is 0.9785. As the test statistics is positive, the null hypothesis is rejected and there is no unit root therefore it is stationary.

Unit root test for Real Price (Industry)

The null hypothesis is that this test has a unit root. The above table shows that the test statistics for industry real price is -0.586 and the p value is 0.8741. As the test statistic is less than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Real Price (Households)

The test statistics for households real price is -0.358 and the p value is 0.9169. The null hypothesis is that this test has a unit root. As the test statistic is smaller than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Unit root test for Income

The test statistics for income is -1.918 and the p value is 0.3236. The null hypothesis is that this test has a unit root. As the test statistic is less than the critical value of the greatest acceptable confidence level (10%), the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary.

Co-integration is a relationship between two non stationary variables; it refers to the fact that these variables share a common stochastic drift and a long term equilibrium relationship. Co-integrated data are never expected to drift too far away from each other and tend to move together in the long run. Engle and Granger formulated a two step process to test for co- integration. The test procedure is to firstly regress one variable (demand) on another variable (income) using ordinary least squares. The second step is to then predict and test a residual for non stationary using the (augmented) Dickey-Fuller unit root test. If the series are co integrated, then the Dickey-Fuller test statistic will be statistically significant. The null hypothesis is that the residual is non stationary. If the residuals are stationary and co-integrated then the null hypothesis will be rejected.

If the variables are co-integrated, they will have a long term or equilibrium relationship between them. As shown in the graphs above, there is a steady upward trend as well as the variability over the years suggesting that these time series are not stationary.

A regression analysis is estimated to produce an equation that will predict a dependent variable from the independent variable and this equation has the form of ln(total demand)=β1 + β2ln(Income) + ut where ut is a white noise error term.

The above stata output shows the analysis of the variance (anova) results located on the top left along with the regression results which is located at the bottom. The dependent variable is demandindustry and the independent variable is income. The coefficients for income and constant are shown in the Coef. column. The Std. Err. column shows the standard error values assosiciated with the coefficients, t is test statistics, P>|t| are the p values, and 95% Confidence Interval for the coefficients. The results from the stata output can be written in regression equation as predicted ln(total demand) =9.782497+0.6714485ln(Income). The coefficient for income is 0.6714485. This leads to a conclusion that an increase in one unit of income, a 0.67144851 unit increase in demand is predicted. The 95% confidence intervals for the coefficients are related to the p values and show the statistical significance of the estimated coefficient. It is significant at 95% level as the p value is less than 0.05, it. The ‘P>|t|’ column shows the two-tailed p-values used to test the null hypothesis that each coefficient is different from 0. Using an alpha of 0.05, the coefficient for income is statistically significant from 0 since the p-value is less than 5%. The constant is an intercept of the regression line and is also significantly different from 0 at the 0.05 alpha level. As the p value (0.000) is smaller than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected. The t-values shows the importance of a variable in the model and also tests whether the coefficient is significantly different from zero. T-values are obtained by dividing the coefficients by its standard error. The t values in this table are relatively large. ‘Prob>F’ is the p-value associated with the F-statistic.  It is used to test whether R-square is different from 0. As it is less than 0.05, there is a statistically significant relationship between demandindustry and income. R-squared is the amount of variance in the dependent variable which can be explained by the independent variable. In this model, R-squared is 0.8318; meaning about 83% of the variance in demand is explained by this model. The adjusted R-squared is a modification of the value of R-squared that adjusts for the number of terms in a model. The adjusted R-squared in this model is 0.8283.

The above stata output shows the analysis of the variance (anova) results located on the top left along with the regression results which is located at the bottom. The dependent variable is demandhousehold and the independent variable is income. The coefficients for income and constant are shown in the Coef. column. The Std. Err. column shows the standard error values assosiciated with the coefficients, t is test statistics, P>|t| are the p values, and 95% Confidence Interval for the coefficients. The results from the stata output can be written in regression equation as predicted ln (total demand) =6.231413+1.361949ln(Income). The coefficient for income is 1.361949. This leads to a conclusion that an increase in one unit of income, a 1.361949 unit increase in demand is predicted. The 95% confidence shows the statistical significance of the estimated coefficient. It is significant at 95% level as the p value is less than 0.05. Using an alpha of 0.05, the coefficient for income is statistically significant from 0 since the p-value of 0.000 is less than 5%. The constant is an intercept of the regression line and is also significantly different from 0 at the 0.05 alpha level. As the p value (0.000) is smaller than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected. T-values are obtained by dividing the coefficients by its standard error. The t values in this table are relatively large. ‘Prob>F’ is the p-value associated with the F-statistic.  It is used to test whether R-square is different from 0. As it is less than 0.05, there is a statistically significant relationship between demandhousehold and income. In this model, R-squared 0.9888; meaning about 99% of the variance in demand is explained by this model. The adjusted R-squared for this model is 0.9886.

The second step in the Engle and Granger’s two step procedure is to test for a unit root. A Dickey Fuller unit root test is conducted on the residual. The null hypothesis is that this test has a unit root. The test statistics for the residual is -2.641 and the p value is 0.0849. As the test statistic is less than the 5% critical value, the null hypothesis is accepted and there is a unit root. It lies within the acceptance region and therefore it is non stationary but the null hypothesis is rejected at the 10% confidence interval.

Error Correction Models are a group of multiple time series models that reconciles the short run behaviour of an economic variable with its long run behaviour. It examines the effect and the speed at which a dependent variable (Y) returns back to equilibrium after a change in an independent variable (X). If two variables are co-integrated, then the relationship between them can be represented as an ECM. It estimates both short and long term effects.

The size of the coefficient value for an independent variable estimates the size of the effect that the variable is having on the dependent variable and the sign on the co-efficient (positive or negative) explains the direction of the effect. The coefficient estimates how much the dependent variable is predicted to increase if the coefficient value is positive or decrease if the coefficient value is negative when the independent variable increases by one holding all other variables constant. The coefficient for the real price industry variable is -0.3171985. For every unit increase in real price for industry, a -0.3171985 unit decrease for demand industry is predicted holding all the other variables constant. There is a significance level of 5%; a p value of 5% or less is the point at which the null hypothesis is rejected. As the p value obtained (0.036) is less than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected that the coefficients are equal to 0. This result concludes that it is stationary and the variables are co-integrated.

The size of the coefficient value for each independent variable estimates the size of the effect that the variable is having on the dependent variable and the sign on the co-efficient (positive or negative) explains the direction of the effect. The coefficient for the real price variable is -0.1272939. For every unit increase in real price for households, a -0.1272939 unit decrease for demand households is predicted holding all the other variables constant. There is a significance level of 5%; a p value of 5% or less is the point at which the null hypothesis is rejected. As the p value obtained (0.216) is less than the significance level (0.05), there is sufficient evidence to reject the null hypothesis therefore the null hypothesis is rejected that the coefficients are equal to 0. This result concludes that it is stationary and the variables are co-integrated.

This graph shows the effect the independent variable (income) has on the dependent variable (demand). There is an upward trend which leads to a conclusion that an increase in a unit of income leads to an increase in a unit of demand for industries.

This graph shows the effect the independent variable (income) has on the dependent variable (demand). There is an upward trend which leads to a conclusion that an increase in a unit of income leads to an increase in a unit of demand for households.

Japan has a population of over 127 million people. In 2011, Japan was ranked third in the world for electricity production after the China and United States with 1,083,142 GWh per year. In terms of daily per capita electricity consumption, the average person in Japan consumed 23.35 kWh compared to 39.25 kWh for an average American. Its per capita electricity consumption increased by 21.8% between 2008 and 2012. Japan was the third largest country in the world with electricity generating capacity of 282 GW in 2010. However after the damage inflicted by the earthquake, the capacity reduced to 243 GW in mid-2011. Japan had 53 active nuclear power generating reactor units in 2009. Almost one quarter (24.93%) of its electricity production was produced from nuclear plants compared to 76.18% for France and 19.66% for the United States. However post earthquake all the nuclear plants were eventually shut down in 2012 and currently produce no nuclear power.

The two above graphs show the relationship between the demand and the real price for electricity in Japan for both industries and households. From 1980 and onwards, the second graph shows a decline in the real prices of electricity, this stimulated the growth of productivity in a wide range of industries. A decrease in real prices led to an increase in demand (shown in graph 1) for electricity for industries and households.