11.2 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

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

Finite Sample Properties for Time Series:

• 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)

11.2.1 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

$$\begin{aligned} y_t &=\beta_0 + \beta_1x_t+\epsilon_t \\ \epsilon_t &= \alpha_1t + v_t \end{aligned}$$

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 partialing 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:

• Feedback Effect

  • $\epsilon_t$ influences next period’s independent variables

• Dynamic Specification

  • include last time period outcome as an explanatory variable

• Dynamically Complete

  • For finite distrusted lag model, the number of lags needs to be absolutely correct.

11.2.2 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)

11.2.3 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

11.2.4 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

• OLS is biased
• Gauss-Markov Theorem
• Finite Sample Properties are invalid

then, we can

• Focus on Large Sample Properties
• Can use [A3a] instead of A3

[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

• Assumptions A1, A2, [A3a]

• [Weak Law] and Central Limit Theorem depend on A5

  • $x_t$ and $\epsilon_t$ are dependent over t
  • without [Weak Law] or Central Limit Theorem depend on A5, we cannot have Large Sample Properties for OLS
  • Instead of A5, we consider [A5a]

• Derivation of the Asymptotic Variance depends on A4

  • time series setting introduces Serial Correlation: $Cov(\epsilon_t, \epsilon_s) \neq 0$

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

11.2.5 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.

2. 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.

11.2.5.0.1 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)