## 18.2 Time Series

$y_t = \beta_0 + x_{t1}\beta_1 + x_{t2}\beta_2 + ... + x_{t(k-1)}\beta_{k-1} + \epsilon_t$

Examples

• Static Model

• $$y_t=\beta_0 + x_1\beta_1 + x_2\beta_2 - x_3\beta_3 - \epsilon_t$$
• Finite Distributed Lag model

• $$y_t=\beta_0 + pe_t\delta_0 + pe_{t-1}\delta_1 +pe_{t-2}\delta_2 + \epsilon_t$$
• Long Run Propensity (LRP) is $$LRP = \delta_0 + \delta_1 + \delta_2$$
• Dynamic Model

• $$GDP_t = \beta_0 + \beta_1GDP_{t-1} - \epsilon_t$$

• A1-A3: OLS is unbiased
• A1-A4: usual standard errors are consistent and Gauss-Markov Theorem holds (OLS is BLUE)
• A1-A6, A6: Finite Sample Wald Test (t-test and F-test) are valid

A3 might not hold under time series setting

• Spurious Time Trend - solvable
• Strict vs Contemporaneous Exogeneity - not solvable

In time series data, there are many processes:

• Autoregressive model of order p: AR(p)
• Moving average model of order q: MA(q)
• Autoregressive model of order p and moving average model of order q: ARMA(p,q)
• Autoregressive conditional heteroskedasticity model of order p: ARCH(p)
• Generalized Autoregressive conditional heteroskedasticity of orders p and q; GARCH(p.q)

### 18.2.1 Deterministic Time trend

Both the dependent and independent variables are trending over time

Spurious Time Series Regression

$y_t = \alpha_0 + t\alpha_1 + v_t$

and x takes the form

$x_t = \lambda_0 + t\lambda_1 + u_t$

• $$\alpha_1 \neq 0$$ and $$\lambda_1 \neq 0$$
• $$v_t$$ and $$u_t$$ are independent
• there is no relationship between $$y_t$$ and $$x_t$$

If we estimate the regression,

$y_t = \beta_0 + x_t\beta_1 + \epsilon_t$

so the true $$\beta_1=0$$

• Inconsistent: $$plim(\hat{\beta}_1)=\frac{\alpha_1}{\lambda_1}$$
• Invalid Inference: $$|t| \to^d \infty$$ for $$H_0: \beta_1=0$$, will always reject the null as $$n \to \infty$$
• Uninformative $$R^2$$: $$plim(R^2) = 1$$ will be able to perfectly predict as $$n \to \infty$$

We can rewrite the equation as

$y_t=\beta_0 + \beta_1x_t+\epsilon_t \\ \epsilon_t = \alpha_1t + v_t$

where $$\beta_0 = \alpha_0$$ and $$\beta_1=0$$. Since $$x_t$$ is a deterministic function of time, $$\epsilon_t$$ is correlated with $$x_t$$ and we have the usual omitted variable bias.
Even when $$y_t$$ and $$x_t$$ are related ($$\beta_1 \neq 0$$) but they are both trending over time, we still get spurious results with the simple regression on $$y_t$$ on $$x_t$$

Solutions to Spurious Trend

1. Include time trend t as an additional control

• consistent parameter estimates and valid inference
2. Detrend both dependent and independent variables and then regress the detrended outcome on detrended independent variables (i.e., regress residuals $$\hat{u}_t$$ on residuals $$\hat{v}_t$$)

• Detrending is the same as partialling out in the Frisch-Waugh-Lovell Theorem

• Could allow for non-linear time trends by including t $$t^2$$, and exp(t)
• Allow for seasonality by including indicators for relevant “seasons” (quarters, months, weeks).

A3 does not hold under:

### 18.2.2 Feedback Effect

$y_t = \beta_0 + x_t\beta_1 + \epsilon_t$

A3

$E(\epsilon_t|\mathbf{X})= E(\epsilon_t| x_1,x_2, ...,x_t,x_{t+1},...,x_T)$

will not equal 0, because $$y_t$$ will likely influence $$x_{t+1},..,x_T$$

• A3 is violated because we require the error to be uncorrelated with all time observation of the independent regressors (strict exogeneity)

### 18.2.3 Dynamic Specification

$y_t = \beta_0 + y_{t-1}\beta_1 + \epsilon_t$

$E(\epsilon_t|\mathbf{X})= E(\epsilon_t| y_1,y_2, ...,y_t,y_{t+1},...,y_T)$

will not equal 0, because $$y_t$$ and $$\epsilon_t$$ are inherently correlated

• A3 is violated because we require the error to be uncorrelated with all time observation of the independent regressors (strict exogeneity)
• Dynamic Specification is not allowed under A3

### 18.2.4 Dynamically Complete

$y_t = \beta_0 + x_t\delta_0 + x_{t-1}\delta_1 + \epsilon_t$

$E(\epsilon_t|\mathbf{X})= E(\epsilon_t| x_1,x_2, ...,x_t,x_{t+1},...,x_T)$

will not equal 0, because if we did not include enough lags, $$x_{t-2}$$ and $$\epsilon_t$$ are correlated

• A3 is violated because we require the error to be uncorrelated with all time observation of the independent regressors (strict exogeneity)
• Can be corrected by including more lags (but when stop? )

Without A3

then, we can

A3a in time series become

$A3a: E(\mathbf{x}_t'\epsilon_t)= 0$

only the regressors in this time period need to be independent from the error in this time period (Contemporaneous Exogeneity)

• $$\epsilon_t$$ can be correlated with $$...,x_{t-2},x_{t-1},x_{t+1}, x_{t+2},...$$
• can have a dynamic specification $$y_t = \beta_0 + y_{t-1}\beta_1 + \epsilon_t$$

Deriving Large Sample Properties for Time Series

under A1, A2, A3a, and A5a, OLS estimator is consistent, and asymptotically normal

### 18.2.5 Highly Persistent Data

If $$y_t, \mathbf{x}_t$$ are not weakly dependent stationary process
* $$y_t$$ and $$y_{t-h}$$ are not almost independent for large h * A5a does not hold and OLS is not consistent and does not have a limiting distribution. * Example + Random Walk $$y_t = y_{t-1} + u_t$$ + Random Walk with a drift: $$y_t = \alpha+ y_{t-1} + u_t$$

Solution First difference is a stationary process

$y_t - y_{t-1} = u_t$

• If $$u_t$$ is a weakly dependent process (also called integrated of order 0) then $$y_t$$ is said to be difference-stationary process (integrated of order 1)
• For regression, if $$\{y_t, \mathbf{x}_t \}$$ are random walks (integrated at order 1), can consistently estimate the first difference equation

\begin{aligned} y_t - y_{t-1} &= (\mathbf{x}_t - \mathbf{x}_{t-1}\beta + \epsilon_t - \epsilon_{t-1}) \\ \Delta y_t &= \Delta \mathbf{x}\beta + \Delta u_t \end{aligned}

Unit Root Test

$y_t = \alpha + \alpha y_{t-1} + u_t$

tests if $$\rho=1$$ (integrated of order 1)

• Under the null $$H_0: \rho = 1$$, OLS is not consistent or asymptotically normal.
• Under the alternative $$H_a: \rho < 1$$, OLS is consistent and asymptotically normal.
• usual t-test is not valid, will need to use the transformed equation to produce a valid test.

Dickey-Fuller Test $\Delta y_t= \alpha + \theta y_{t-1} + v_t$ where $$\theta = \rho -1$$

• $$H_0: \theta = 0$$ and $$H_a: \theta < 0$$
• Under the null, $$\Delta y_t$$ is weakly dependent but $$y_{t-1}$$ is not.
• Dickey and Fuller derived the non-normal asymptotic distribution. If you reject the null then $$y_t$$ is not a random walk.

Concerns with the standard Dickey Fuller Test
1. Only considers a fairly simplistic dynamic relationship

$\Delta y_t = \alpha + \theta y_{t-1} + \gamma_1 \Delta_{t-1} + ..+ \gamma_p \Delta_{t-p} +v_t$

• with one additional lag, under the null $$\Delta_{y_t}$$ is an AR(1) process and under the alternative $$y_t$$ is an AR(2) process.
• Solution: include lags of $$\Delta_{y_t}$$ as controls.
1. Does not allow for time trend $\Delta y_t = \alpha + \theta y_{t-1} + \delta t + v_t$
• allows $$y_t$$ to have a quadratic relationship with t
• Solution: include time trend (changes the critical values).

Adjusted Dickey-Fuller Test $\Delta y_t = \alpha + \theta y_{t-1} + \delta t + \gamma_1 \Delta y_{t-1} + ... + \gamma_p \Delta y_{t-p} + v_t$ where $$\theta = 1 - \rho$$

• $$H_0: \theta_1 = 0$$ and $$H_a: \theta_1 < 0$$
• Under the null, $$\Delta y_t$$ is weakly dependent but $$y_{t-1}$$ is not
• Critical values are different with the time trend, if you reject the null then $$y_t$$ is not a random walk.
##### 18.2.5.0.1 Newey West Standard Errors

If A4 does not hold, we can use Newey West Standard Errors (HAC - Heteroskedasticity Autocorrelation Consistent)

$\hat{B} = T^{-1} \sum_{t=1}^{T} e_t^2 \mathbf{x'_tx_t} + \sum_{h=1}^{g}(1-\frac{h}{g+1})T^{-1}\sum_{t=h+1}^{T} e_t e_{t-h}(\mathbf{x_t'x_{t-h}+ x_{t-h}'x_t})$

• estimates the covariances up to a distance g part

• downweights to insure $$\hat{B}$$ is PSD

• How to choose g:

• For yearly data: g = 1 or 2 is likely to account for most of the correlation
• For quarterly or monthly data: g should be larger (g = 4 or 8 for quarterly and g = 12 or 14 for monthly)
• can also take integer part of $$4(T/100)^{2/9}$$ or integer part of $$T^{1/4}$$

Testing for Serial Correlation

1. Run OLS regression of $$y_t$$ on $$\mathbf{x_t}$$ and obtain residuals $$e_t$$

2. Run OLS regression of $$e_t$$ on $$\mathbf{x}_t, e_{t-1}$$ and test whether coefficient on $$e_{t-1}$$ is significant.

3. Reject the null of no serial correlation if the coefficient is significant at the 5% level.

• Test using heteroskedastic robust standard errors
• can include $$e_{t-2},e_{t-3},..$$ in step 2 to test for higher order serial correlation (t-test would now be an F-test of joint significance)