Time Series Analysis - Reading 6

Question 1

What is a time series?

Answer 1

is a set of observations for a variable over successive periods of time

Question 2

What is a linear trend and how to represent it?

Answer 2

A linear trend is time series pattern that can be graphed using a straight line. The simplest form of linear trend is represented by the following linear trend model :
y_t=b_0+b_1(t)+e_t
*via OLS

Question 3

How can a time series that exhibit a exponential growth can be modeled?

Answer 3

y_t=e^[b_0+b_1(t)]

ln(y_t)=b_0+b_1(t)

Question 4

Why should s linear trend should be used?

Answer 4

around the trend a time series that best modeled with a log linear trend model ratter than a linear trend model

Question 5

What is the limitations of. trend models?

Answer 5

One of the assumptions underlying liner regression is that the residuals are uncorrelated with each other. A violation of this assumption is referred to as autocorrelation -> May be the case that even a log- linear model is not appropriate
in the presence of serial correlation. In this case, we will want to turn an autoregressive model (AR).
*DW to test for serial correlation

Question 6

What is that will make statistical inferences valid about a time series model?

Answer 6

covariance stationary

Question 7

What makes a time series covariance stationary?

Answer 7

1. Constant and finite expected value
2. Constant and finite variance
3. Constant and finite covariance between values of any given lag

Question 8

What happens when a AR model exhibit serial correlation?

Answer 8

The AR model is not correctly specified. When the error terms are correlated, standard errors are unreliable
and t- tests of individual coefficients can incorrectly show statistical significance or
insignificance

Question 9

Steps for specyinfing an AR model?

Answer 9

1. Estimate the AR model being evaluated using linear regression
2. Calculate the autocorrelations of the model’s residuals
   3, Test whether the autocorrelation are significantly different from zero
   *standard error= 1/(T)^0,5
   *t=autocorrelation/1/(T)^0,5 with N-2 df

Question 10

What is a mean reverting level?

Answer 10

A time serie exhibls MEAN REVERSION if it has
a tendency to move toward is mean. In other words the time series has a tendency to decline when the current- value is above the mean and
rise when the current value is below the mean. If a time series is at its mean - reverting level, the model predicts that the next value of
the time series will be the same as its current value.
x_t = b_0/(1-b_1)

*all covariance stationary series have finite mean-reverting level, lag coefficient less than 1

Question 11

What are in sample forecast?

Answer 11

In -sample forecasts are within the range of data used to estimate the model, which for a time series is known
as the sample or leak period. In-sample forecast errors are, where t is an observation. within the sample period.

Question 12

What are out-of sample forecast?

Answer 12

Out-of-sample forecasts are made outside of the sample period. In other words, we compare how accurate
a model is in foresting the y variable value for a time period outside the period used lo develop the model. Out-of-sample forecasts are important because they provide a test whether the model adequately describes the time series and whether if has relevance in the real world.

Question 13

For what is Root Mean Squared Error is used?

Answer 13

The Root Mean Squared Error criterion (RMSE) is used to compare the accuracy of auto regressive modals in fore
casting out-of-sampie values.

Tha model with the low RMSE for the out-of-sample data will have lower forecast error and will be expect to have better
predictive power in the future.

Question 14

What is the tradeoff between shorter time periods and long time periods in financial data?

Answer 14

Models estimated with shorter time series are usually more stable than those with longer
time series because a longer time sample period increases the chance that the underlying process has changed
tradeoff between incremental statistical reliability

Question 15

What is a random walk?

Answer 15

It a time series follows a random walk process the predicted value of the time series in one period is equal; to the value of the series in the previous period plus a random error term

Question 16

What is a random walk with a drift?

Answer 16

If a time series follows a random walk w a drift the intercept term is not equal to zero

Question 17

Is a random walk with or without a drift a stationary process?

Answer 17

No, because it has not a finite mean reverting level

Question 18

What is having a unit root mean?

Answer 18

If the value of the lag coefficient is equal to one, the time series is said to have a unit root and will follow a random walk process.

Question 19

How to test if a series is covariance stationary?

Answer 19

1. Run an AR model and examine autocorrelation

2. Perform a DF Test

Question 20

How a dickey Fuller test work?

Answer 20

1. Start w/ the basic form of the AR(1) model
2. Subtract X_t-1 from both sides
   If (b_1-1) is not significantly different from zero they say that b_1 must be equal to one and, therefore, the series must be a unit root

Question 21

How to fix a series that has a unit root process?

Answer 21

First Differencing
Covariance stationarity can often be achieved by transforming the data using first differencing and modeling the first-differenced time series as an autoregressive time series.

Question 22

What is seasonality?

Answer 22

Seasanaliyy in a time-series is a pattern that tends do repeat from year to year

Question 23

How to adjust a model for seasonality?

Answer 23

To adjust for seasonality in an AR model, an additional lag of the dependent variable (corresponding to the
same period in the previous year

Question 24

What is an ARCH model?

Answer 24

ARCH exists if the variance of the residuals in one period is dependent on variance of the residuals in a previous period. When this condition exists, the standard errors of the regression coefficients in AR, models and the hypothesis tests of these coefficients are invalid.

Question 25

How to test a time series to see if it is an ARCH model?

Answer 25

ARCH(1):
e^_t=a_0+a_1(e^_t-1)
if a_1 is statistically significant then the series is ARCH(1)

Question 26

What is cointegration?

Answer 26

Cointegration means that two time series are economically linked or follow the same trend and that relationship is
not expected to change. If two time series are cointegrated, the error term from regressing one on the other is covariance stationary and the t-tests are reliable. This means that scenarie 5 will produce reliable regression estimate, wheres
scenario 4 will not.

4. Neither time series is covariance stationary and the two series are not cointegrated.
5. Neither time series is covariance stationary and the two series are cointegrated.

Question 27

Given: y_t=b_0+b_1x_t+e_t

Must run a separate DF test with 5 possible results?

Answer 27

1. Both time series are covariance stationary. ->RELIABLE
2. Only the dependent variable time series is covariance stationary.->NOT RELIABLE
3. Only the independent variable time series is covariance stationary.->NOT RELIABLE
4. Neither time series is covariance stationary and the two series are not cointegrated.->NOT RELIABLE
5. Neither time series is covariance stationary and the two series are cointegrated.->RELIABLE