Reading 13: Time Series

Question 1

What is a Time series?

Answer 1

A set of obsrvations on a variable’s outcomes in different time periods.

(quarterly sales for example)

Question 2

What are the challenges in working with time series?

Answer 2

Often with time series the assumptions of a linear regression model are not satisfied.

What is not satisfied?

1. the residual errors are correlated instead of uncorrelated
2. the mean and/or variance of the time series changes over time.

Question 3

What type of Time Series are there?

Answer 3

1. Trend Models

• Linear Trend Models
• Log-Linear Trend Models

2. Lagged Models
   * Autoregressive Model (AR)

Question 4

What is a linear Trend Model

Answer 4

A linear trend model is using time to explain/forecast the dependent variable

Question 5

What are Log-Linear Trend models?

Answer 5

A log Linear Trend Model is a model that has a dependent variable that grows at a constant rate and hence the formula needs to be adjusted for that by taking the Natural Log of the Dependent variable.

Question 6

How do we test for correlation errors in a Trend Model?

Answer 6

We use the Durbin Watson test on the Residuals. This test is the Test with the D_l and D_u by looing up the values in the table

DW = 2(1-R)

Question 7

What are Autoregressive (AR) time-series Models?

Answer 7

An Autoregressive Model (AR) is a time series regressed on its own past values, and represent this relationhsip effectively

Question 8

What is covariance stationary and why is it nessecary?

Answer 8

Covariance stationary basically means that only in a state in which a time series has:

• constatnt and finite expected values
• constant and finite variance
• constant and finite covariance (with leading or lagged values)

will the resulting regression be meaningful.

Question 9

Why is serial correlation in an AR model a problem?

Answer 9

If serial correlation exists, this means that the model does not include all the information out of the data yet.

Solution: Add more lags in the model

Question 10

How do we test for Serial Correlation in an AR model?

Answer 10

We CANNOT use the Durbin-Watson test in an AR model

We need to use the T-Tset, where:

• Standard error = 1/ SqRoot(T)
• Use a t-stat with df = t-1 for samples
• With level of significance (Alpha)

We look at the autocorrelation within the residuals!

Question 11

What do you do if there are significant readings in the t-statistics in the autocorrelation of the residuals?

Answer 11

You add antoher order!

Rerun the statistics and check for serial correlation

Question 12

What is Mean Reversion?

Answer 12

Mean reversion in a time series means that if it tends to fall when its above its mean and rise when it its level is below its mean.

Formula mean Reversion: x_t= b₀ / 1-b₁

Question 13

How do we test for the accurracy of the AR model?

Answer 13

This is measured by the Root Mean Squared Error (RMSE)

= Square Root of the average squared error

= SqRoot (MSE)

Question 14

What is a random Walk in a time series?

Answer 14

This means that there is no tendency to revert back to its mean level in the next period.

If this follows a random pattern, we call it a random walk.

= Value of X = value of X_t-1 + error

The error is unpredictable and random.

Question 15

What type of Random walks are there?

Answer 15

1. No Drift Random Walk

X_t = 0 + (1)X_t-1 + error

when No drift b₁ = 1

2. Random walk with a drift

Xt = b₀ + X_t-1 + error

Question 16

What is First Differencing?

Answer 16

creating of a new dependent variable in order to overcome problems with random walks in a time series (AR)

Question 17

What is an ARCH model?

Answer 17

An ARCH model is a model that is an Autoregressive Conditional Heteroskedasticity.

Means non-stationarity in multiple time series.

• Variance is not constant
• Variance in one time period is correlated with the variance of the residulas in another
• The standar errors of the coefficients in AR models are unreliable

-> Hyphotesis testing might be wrong!

Question 18

How to test for ARCH?

How to Solve it?

Answer 18

Squared residulas are regressed on the first lag of the squared residuals:

error_²t = a₀+ a₁ error²_t-1 + µ_t

If a1 = significant -> ARCH positive

If so: Solve by GLS (Generalized Least Squares)

Question 19

In an ARCH model, can you predict volatitlity and if so how?

Answer 19

Yes if an ARCH model is significant and there is Heteroskedasticity, you can use the formula below to forcast volatility of the coming period

σ²_t+1= a₀ + a₁ error²_t

Question 20

What is Cointegration?

Answer 20

Cointegration is when Two time series are related to the same macro variables or follow the same trend.

If the model is cointegrated -> model can be used

If the model is not cointegrated -> NOT GOOD -> throw out model!