Inference with time series

OLS on time series data

Bias of static models and spurious correlations

Static models

Static models are of the form:

$$y_{t}=\alpha+\beta x_{t}+\epsilon_{t}$$

These have no lagged variables or difference operators.

Time series

Taking differences

What we use should depend on I(1), I(0) etc from ADF

if we’re missing time invariant data, we can do first differences and this isn’t a problem if we do diff in diff this removes trends?

page on first difference estiamtion? OLS on first differences. No other lags page on first difference ESTIMATOR

Discontinuity

Create a dummy for before/after a date.

Panel data

Difference-in-difference

Consider the grouped linear model:

$$y_{ij}=\mu+\tau_i+X_{j}\theta +\epsilon_{ij}$$

By taking differences with another observation in the same group we remove the average terms.

$$y_{ij}-y_{ik}=(\mu+\tau_i+X_{j}\theta +\epsilon_{ij})- (\mu + \tau_i + X_{k}\theta + \epsilon_{ik})$$

$$y_{ij}-y_{ik}=(X_{j}\theta - X_{k})+(\epsilon_{ij}- \epsilon_{ik})$$

diff in diff: control group and treated group. page on leakiness? are control affected too? Assumption: in absense of treatment, price would have evolved like control

Structural breaks

Testing for structural breaks with the Chow test.

Dynamic or lagged independent variables

Static panel data: No lags of independent variables. Dynamic panel data: Lags of independent variables.

OLS is consistent for static panel data, not for dynamic This results in Nickell’s bias for dynamic panel data

Dynamic panel data: $$y_{t-1}$$ is a regressor Panel data estimation: LSDV. Least squares dummy variable estimator arnello bond