TIME SERIES

Time Series Analysis

[Name of the Institute]

Question no. 1

Graph of Spot Prices and Lag of Future Prices

Future prices of a commodity are always expected to be higher due to inflation. The difference between a commodity's price in one time-period and the expected future price in the previous time period is actually rise in price levels. The graph above is a plot of spot prices of oil and the lag of future prices for daily 5-days data from 1988 to 2012. The prices are in U.S. dollar. For first two decades of the total time-period the spot prices and future prices fluctuated from $10 to $40 per barrel. For the same time-period, the increase in price level per day is almost constant. In the decade of 2000 both spot and future oil prices increased drastically. The highest price of which reached to $149 per barrel in 2008. Even then the difference between the spot and future prices per day remained unchanged. This represents that the high prices were anticipated. The year 2012 has a different pattern as the difference between the spot price and previous time's future price is increasing implying that the prices of oil are unanticipated.

Augmented Dickey-Fuller

Augmented Dickey-Fuller is a test to check whether a series is stationary or non-stationary, or if a series possesses unit root. The test uses following regression to determine the existence of unit root in a time-series variable (Yaffe & McGee, 200, p. 84). It often happens while dealing with time-series data that the mean and variances of a variable are not constant. This situation is problematic if the objective of data analysis is prediction. There exists a unit root if a time-series variable is non-stationary.

Sometimes the unit root vanishes when the values are subtracted from past values of the variable. This is called first difference. If a time-series variable has no unit root it is called an integrated series of order zero, represented as I(0). If the unit root exists at level form but vanishes by taking first difference, the series is called integrated of order 1 I(1). Similarly, if the unit root exist at level form and at first difference but vanishes at second difference the series is called integrated of order two I(2). Following are the results of ADFL for spot prices and future prices of oil.

t-stat

Level

1st Difference

2nd Difference

Spot Prices

-0.201

-13.08

-11.20

Future Prices

-0.56

-19.36

-18.59

Unit Root

Exist

Does not exist

Does not exist

Above results are obtained using the unit-root test available in E-views namely Augmented Dickey-Fuller. The test in E-views takes number of lags according to SBIC. Both the variables in given regression is non-stationary at level form, and stationary at 1st difference and 2nd difference. Thus, we can obtain cointegeration of the regression.

Spurious Regression

A regression might be a spurious regression if there exists individual significance of the variables but the overall significance is low. An example is an OLS regression result where t-statistics are significant but R2 is very low (Gujarati, 2007, p. 806). One reason of spurious regression is non-stationary time-series data ...