The Unit Root Test for Nonstationary
Unit root testing involves checking whether the time series is covariance stationary. We can either form an AR model and check for autocorrelations or perform a Dickey and Fuller test. A t-test is performed to examine the statistical significance of…
Unit Roots for Time-Series Analysis
The Unit Root Problem An AR(1) series is said to be covariance stationary if the absolute value of the lag coefficient \(\text{b}_{1}\) is less than 1. If the absolute value of \(\text{b}_{1}=1\), the time series is said to have a…
Random Walk Process
A time series is said to follow a random walk process if the predicted value of the series in one period is equivalent to the value of the series in the previous period plus a random error. A simple random…
Coefficient Instability
Time series coefficient estimates can change over time. Regression coefficient estimates derived from an earlier sample period can differ from those approximated using a later period. Therefore, sample period selection is crucial in estimating valuable models. As a result, different…
Multiperiod Forecasts
In-sample Forecasts An in-sample forecast uses the fitted model to derive the predicted values within the period used to estimate model parameters. In-sample forecast errors are residuals generated from a fitted-time series model. For instance, if we use a linear…
The Mean Reversion
Mean reversion refers to the behavior of a time series to fall when its values are above the mean and rise when they are below the mean. This is illustrated as follows: A mean-reverting time series tends to move towards…
Residual Autocorrelation
The autocorrelation of a time series refers to the correlation of that time series with its past values. The kth order autocorrelation is the autocorrelation between one time series observation and the value k periods before. We cannot use the…
Autoregressive Models and Multiperiod Forecasts
The current-time values of a time series are related to the previous time values. This property is termed autoregressive. Autoregressive models are abbreviated (\(AR_{p}\)) models. \(p\) is known as the order of the model. It indicates the number of lagged…
Covariance Stationary Property
A time series is said to be covariance stationary if its properties, such as the mean and variance, remain constant over time. A time series that is nonstationary leads to invalid linear regression estimates with no economic meaning. A time…
Linear or Log-Linear Model
A linear trend model should model a time series that increases over time by a constant amount. On the other hand, a time series that grows at a constant rate should be modeled by a log-linear model. To decide between…
