# 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 models are best suited for different periods. Using only one model for the entire period can result in a poorly fitting model.

Similarly, the estimated regression coefficients vary depending on the length of the sample periods. The choice of the model to be used in the analysis depends on the sample period. An AR(1) model may be best suited for a single period, but an AR(2) model may fit a previous or later period better. When the two periods are combined, we are likely to select either the AR(1) or AR(2) model for the combined period. This implies that at least a one-time span of the data will be fitted poorly.

Moreover, a longer period increases the risk of mean and variance being unstable over time. Conversely, using a shorter period could produce insufficient data, resulting in lower confidence in estimated parameters.

There are no definitive guidelines for determining the appropriate sample period. However, some basic guidelines are recommended, including:

• Use basic sampling theory: Avoid using two distinctly different populations.
• Consider basic time-series properties: Do not combine stationary and nonstationary series. Additionally, do not mix series with different mean and variance terms.
• The length of the sample period: A longer sample period implies that the sample probably comes from different populations.

## Question

Which of the following statements is most accurate?

1. Coefficients of models estimated with longer time series are usually more stable than those with shorter time series.
2. Using a shorter period to estimate coefficients results in higher confidence in estimated parameters.
3. Coefficients of models estimated with shorter time series are usually more stable than those with longer time series

#### Solution

The correct answer is C.

Models estimated with shorter time series are usually more stable than those with longer time series. A longer sample period increases the risk of the mean and variance being unstable over time.

A is incorrect. Models estimated with longer time series are usually unstable relative to those with shorter time series because a longer sample period increases the likelihood that the underlying economic process has changed.

B is incorrect. Using a shorter period could produce insufficient data, resulting in lower confidence in estimated parameters.

Reading 5: Time Series Analysis

LOS 5 (h) Explain the instability of coefficients of time-series models.

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