Statistics Chapter 9: Time-Series Models

Study Notes

Time series data tracks an observation unit (e.g., individual, country, firm) over multiple periods. Correlation between data points and the order of observations are key features not found in cross-sectional data.
Time-series models require assumptions beyond linear regression, focusing on the error term's distribution remaining consistent over time. This includes constant mean and variance, and constant correlation between error terms across periods.
R package dynlm is used for time series models, including lags and specific operators.
R package zoo handles irregular or high-frequency time series. Function ts() structures data as time series in R, specifying start date, end date, and frequency (annual, quarterly, monthly).

A finite distributed lag model (FDL) represents a linear relationship between a dependent variable (y) and lagged values of an independent variable (x). The order (q) denotes the number of lags.
ẞ_s represents the multiplier for an s-period delay.
ẞ₀ is the immediate impact of x on y.
The interim multiplier is the total change in y after s periods if x increases by one unit.
The total multiplier sums all ẞ_s values.
Okun's law is an example using unemployment and GDP growth, considering changes in unemployment rate as a function of lagged growth rates.

Serial correlation (autocorrelation) measures the correlation between observations separated by certain periods in a time series.
Autocorrelation is a key difference between time series and cross-sectional data.
Autocorrelation is assessed via correlograms, which are bar graphs visually representing √Tr_k (the test statistic for lag k). If the bars exceed confidence intervals, it indicates autocorrelation. The correlogram visualizes the correlation values between pairs of observations at different lags, starting with lag 0 (representing contemporaneous correlation).

Nonlinear least squares estimation is needed for models where the relationships are not linear (e.g., incorporating AR(1) errors).
This method uses the nls() function in R for fitting models with nonlinear components.

Forecasting uses a model to predict future values of a time series variable.
Techniques like autoregressive (AR) methods for forecasting (using lags of the variable only) and exponential smoothing (weighted average of past values).
ARDL models combine both autoregressive features with distributed lag components.
The choice of forecasting method often hinges upon the model characteristics and aims.

Cointegration describes a "long-run" relationship between two nonstationary time series, where the differences/residuals are stationary.
The po_test() function helps identifying cointegration through stationarity tests on the residuals.
dynlm() can be used to test for cointegration by regressing one series on the other.

ECMs combine short-term and long-term effects in cointegrated relationships.
ECMs incorporate lagged differences and often account for more lagged variables within the model.
The nls() function is used for fitting and analyzing error correction models.