Time-series

Linear Trend Model

Work with constant change amount:

y_{t} = b_{0} + b_{1} t + ϵ

Log-Linear Model

Applying log transform to work with constant growth rate time series:

y_{t} = e^{b_{0} + b_{1} t + ϵ}

The trend models are generally not appropriate for time series when data are serial correlated. We can use Durhin-Watson test to detect serial correlation.

Autoregressive Model (AR)

Autoregressive model use past values of dependent variables as independent variables.

A R (p) : x_{t} = b_{0} + b_{1} x_{t - 1} + \dots + b_{p} x_{t - p} + ϵ

Additional assumption: Covariance Stationary:
- Constant and finite expected value in all periods
- Constant and finite variance in all periods
- Constant and finite covariance with itself for a fixed number of periods among all periods.

Detecting Serial Autocorrelation

Fit the $A R (1)$ model
Compute autocorrelations
View ACF plot and decide.
t test: $t = \frac{\hat{ρ_{ϵ_{t}, ϵ_{t - k}}}}{1 / \sqrt{T}}$ , $H_{0} :$ No autocorrelation.

Moving Average Model (MA)

Moving Average model use lagged residuals to model:

x_{t} = ϵ_{t} + θ ϵ_{t - 1}

For an $M A (q)$ model the first $q$ autocorrelations will be significantly different from $0$ . All beyond that will be equal to $0$ .

Comining will give ARIMA model.

Violations of Assumptions

Seasonality

Seasonality refers to time series show regular patterns of movement within the year. It is to include a seasonal lag in AR model.

Unit Root

Mean Reversion

A time series shows mean reversion. if it tends to move to its mean.

For an $A R (1)$ model, its mean-reverting level is $x_{t} = \frac{b_{0}}{1 - b_{1}}$ , when $| b_{1} | < 1$ and has covariance stationary.

NOTE

When $b_{1} = 1$ , $A R (1)$ becomes random walk. If $b_{0} = 0$ it is simple random walk, if not, it is random walk with drift.

Random walt will not exhibit covariance stationary. In this case, least square method cannot be trusted to model $A R (1)$ . In this case, we say the time series has a unit root.But t-test cannot be used to test unit root.

Therefore, Dicky-Fuller test will bne used to test unit root. We test $x_{t} - x_{t - 1} = b_{0} + (b_{1} - 1) x_{t - 1} + ϵ_{t}$ . We test if $g_{1} = b_{1} - 1 = 0$ :

$H_{0} : g_{1} = 0, H_{1} : g_{1} < 0$ . We calculate $t$ statistics and use revised critical value (larger than usual).

First Differencing

Applying first order differencing on random walk will result in $y_{t} = x_{t} - x_{t - 1} = ϵ_{t}$ , which stabilize the time series.

Heteroskedasticity

Recall: Conditional heteroskedasticity defines case where the heteroskedasticity of the error variance is correlated with the values of the independent variables.

When Conditional heteroskedasticity exists in AR model, we call it ARCH.

ARCH Model

In addition to $A R$ , $A R C H$ also model the variance of residuals:

σ_{ϵ_{t}}^{2} = a_{0} + a_{1} ϵ_{t - 1}^{2} + \dots + u_{t}

$G A R C H$ also include past variance of residuals to model (e.g $σ_{ϵ_{t - 1}^{2}}$ )

NOTE

We should use Generalized Least Square to fit the model if conditional heteroskedasticity exists.

Regression With More Than One Time Series

Test unit root for both
If both have unit root. If they have cointegration (long-term financial or economic relationship so that they do not diverge from each other in the long run), we can use linear regression
To test cointegration, we apply Engle and Granger test to test unit root on the residual term.We want to make sure error term has no unit root.

Time-series ​

Linear Trend Model ​

Log-Linear Model ​

Autoregressive Model (AR) ​

Detecting Serial Autocorrelation ​

Moving Average Model (MA) ​

Violations of Assumptions ​

Seasonality ​

Unit Root ​

Mean Reversion ​

First Differencing ​

Heteroskedasticity ​

ARCH Model ​

Regression With More Than One Time Series ​

Model Selection ​