Time Series Notes
Regression with Time Series Data Week 10
Main features of Time series Data
Observations have temporal ordering
Variables may have serial correlation, trends and seasonality
Time series data are not a random sample because the observations in time series are collected from the same objects at different points in time
For time series data, because MLR2 does not hold, the inference tools are valid under a set of strong assumptions (TS1-6) for finite samples
While TS3-6 are often too restrictive, they can be relaxed for large samples. In essence, TS1, TS2, (z10), (h10) and (u10) are sufficient for large sample inference
Serial correlation of a time series variable is the correlation between the variable at one point in time with the same variable at another point in time
(z10), (h10), (u10) z10 = E(ut | xt) = 0
When (z10) holds then the regressors are contemporaneously exogenous and OLS is consistent but is not sufficient for OLS to be unbiased
When TS3 holds, which implies (z10), then the regressors are strictly exogenous and OLS is unbiased h10 = Var(ut | xt) = 2 and is known as contemporaneous homoskedasticity and is a weaker assumption than TS4 u10 = E(utus | xt,xs) = 0 and is a weaker assumption than TS5
FDL model and LRP
A FDL model allows the lags of one or more variables to affect the dependent variable
The LRP is the impact on y of a permanent one unit shift in z at t
Trends and Seasonality
Seasonal dummy variables can be used to account for seasonality with the first quarter as base and define three dummy variables
Trends can be accounted for by adding a time trend in the model
Estimate B1 and B2 by regressing yt, xt1, xt2 on the time trend and seasonal dummies and save the residuals respectively. Regressing the residual of yt on the two residuals of xt1 and xt2 without intercept.
Further Issues with Time Series Data Summary Week 11
1. Asymptotic properties of OLS with time series
Theorem 11.1 (consistency)
Under TS1-2 and (z10), if
Main features of Time series Data
Observations have temporal ordering
Variables may have serial correlation, trends and seasonality
Time series data are not a random sample because the observations in time series are collected from the same objects at different points in time
For time series data, because MLR2 does not hold, the inference tools are valid under a set of strong assumptions (TS1-6) for finite samples
While TS3-6 are often too restrictive, they can be relaxed for large samples. In essence, TS1, TS2, (z10), (h10) and (u10) are sufficient for large sample inference
Serial correlation of a time series variable is the correlation between the variable at one point in time with the same variable at another point in time
(z10), (h10), (u10) z10 = E(ut | xt) = 0
When (z10) holds then the regressors are contemporaneously exogenous and OLS is consistent but is not sufficient for OLS to be unbiased
When TS3 holds, which implies (z10), then the regressors are strictly exogenous and OLS is unbiased h10 = Var(ut | xt) = 2 and is known as contemporaneous homoskedasticity and is a weaker assumption than TS4 u10 = E(utus | xt,xs) = 0 and is a weaker assumption than TS5
FDL model and LRP
A FDL model allows the lags of one or more variables to affect the dependent variable
The LRP is the impact on y of a permanent one unit shift in z at t
Trends and Seasonality
Seasonal dummy variables can be used to account for seasonality with the first quarter as base and define three dummy variables
Trends can be accounted for by adding a time trend in the model
Estimate B1 and B2 by regressing yt, xt1, xt2 on the time trend and seasonal dummies and save the residuals respectively. Regressing the residual of yt on the two residuals of xt1 and xt2 without intercept.
Further Issues with Time Series Data Summary Week 11
1. Asymptotic properties of OLS with time series
Theorem 11.1 (consistency)
Under TS1-2 and (z10), if