# 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