Time-Series Analysis

Question 1

Autoregressive AR model

Answer 1

A time series regressed on its own past values.

A statistical model is autoregressive if it predicts future values based on past values. For example, an autoregressive model might seek to predict a stock’s future prices based on its past performance.

Question 2

DW test for Serial correlation in linear/log-linear model hypothesis

Answer 2

H0: Dw = 2 - Fail to reject - Do not reject the null hypothesis - No Serial correlation
Ha: Dw =/2 -Reject null - We have serial correlation

Question 3

What are the 3 properties we must satisfy to have “Covariance Stationary Series”

Answer 3

Mean, Variance, and Cov(yt, yt-s) must be constant and finite in all periods.

1. The expected value of the time series must be constant and finite in all periods.
2. The variance of the time series must be constant and finite in all periods.
3. The covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all period

Question 4

What is “mean Reversion”

Answer 4

The value of the time series falls when it’s above its mean, and rises when it’s below its mean.

Mean reversion in finance suggests that various relevant phenomena such as asset prices and volatility of returns eventually revert to their long-term average levels.

The mean reversion theory has led to many investment strategies, from stock trading techniques to options pricing models.

Mean reversion trading tries to capitalize on extreme changes in the price of a particular security, assuming that it will revert to its previous state

Question 5

Define the Mean reverting level …

Xt > b0/(1-b1)

Answer 5

The time series will decrease

Question 6

Define the Mean reverting level …

Xt = b0/(1-b1)

Answer 6

The time series will remain the same

Question 7

Define the Mean reverting level …

Xt < b0/(1-b1)

Answer 7

The time series will remain the increase

Question 8

What is an “in-sample forecast”

Answer 8

Prediction
Predicted vs Observed values to generate the model

Models with a smaller variance of errors are more accurate

Question 9

What is an “out-of-sample forecast”

Answer 9

Forecast
Forecast vs Outside the model’s values

Use Root Mean Squared Errors (RMSE) - used to compute out-of-sample forecasting performance. The smaller the RMSE, the better.

Question 10

hat 2 elements does Random Walk not have?

Answer 10

Finite mean reverting level, and finite variance

Question 11

Which test do we use to test for unit root?

Answer 11

Dickey-Fuller test

Question 12

When testing for Unit root
If the coefficient is |b1| < 1

Answer 12

No unit root - the time series is covariance stationary

Question 13

When testing for Unit root
If the coefficient is b1 = 1

Answer 13

Unit root.
Time series is a random walk.
It is not covaraince stationary

Question 14

DW Test for SC

Result from model output (DW statistics) < DW Critical

Answer 14

Evidence of Positive Serial Correltion
we can reject the hypothesis of no Positive Serial correlation

Question 15

DW Test for SC
Result from model output (DW statistics) > DW Critical

Answer 15

NO Evidence of Positive Serial Correltion

Question 16

When are residuals serially correlated in AR model test statistics?

Answer 16

|T-Stat| < Critical Value

Question 17

When are residuals are not serially correlated in AR model test statistics?

Answer 17

|T-Stat| > Critical Value

Question 18

The standard error of the autocorrelations is calculated as…

Answer 18

1/√T

where T represents the number of observations used in the regression

Question 19

Explain the DW test

Answer 19

The DW statistic is designed to detect positive serial correlation of the errors of a regression equation.

Under the null hypothesis of no positive serial correlation, the DW statistic is 2.0.

Positive serial correlation will lead to a DW statistic that is less than 2.0.

We do NOT want positive serial correlation !!!

Question 20

The steps to calculate RMSE …

Answer 20

The steps to calculate RMSE are as follows:

1. Take the difference between the actual and the forecast values. This is the error.
2. Square the error.
3. Sum the squared errors.
4. Divide by the number of forecasts.
5. Take the square root of the average

Question 21

What is Root Mean Squared Error (RMSE)

Answer 21

A model’s accuracy in forecasting out-of-sample values is assessed using the root mean squared error (RMSE).

RMSE is the square root of the mean squared error. The model with the smallest RMSE is seen as the most accurate, as it is perceived to have better predictive power in the future.

Question 22

What is a unit root

Answer 22

Is a stochastic trend in a time series

Random Walk with a drift

If TS has unit root, it shows a systamatic pattern that is unpredicable

Question 23

How do we transform TS into covariance stationary

Answer 23

By using first differencing

Regression 2:yt = b0 + b1yt−1 + εt,
where yt = xt − xt−1.

Question 24

Can we test for positive SC if we have lag variables using DW test?

Answer 24

NO!!!

DW can be used for linear models, not trend models

Question 25

When testing for serial correlation using DW test.

Answer 25

0 ——-|dl|——–|du|——-2
Between 0 and lower level = + SC
Between Du and 2 = Okay
Between Dl and Du = We don’t know

Question 26

Adjusted R square formula

Answer 26

1 - [(n-1)/(n-k-1)] x (1-R Squared)