##### Choosing the Appropriate Time-Series Model

The following guidelines are used to determine the most appropriate model depending on the need: Understand the investment problem. This is followed by choosing the initial model. Plot the time series to check for covariance stationarity. Observe if there is…

##### Cointegration

Consider a time series of the inflation rate \((\text{y}_{\text{t}})\) regressed on a time series of interest rates \((\text{x}_{\text{t}})\): $$\text{y}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}}+\epsilon_{\text{t}}$$ In this case, we have two different time series, \(\text{y}_{\text{t}}\) and \(\text{x}_{\text{t}}\). Either one of the time series is subject to…

##### Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is the dependence of the variance of the error term on the independent variable. We have been assuming that time series follows the homoskedasticity assumption. Homoskedasticity is the independence of the variance of the error term on the independent…

##### Seasonality

Seasonality is a time series feature in which data shows regular and predictable patterns that recur every year. For example, retail sales tend to peak for the Christmas season and then decline after the holidays. A seasonal lag is the…

##### 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…