Univariate Time-series

Question 1

Define autocovariance

Answer 1

Autocovariance_s = E[(y_t - mu) (y_(t-s) - mu)]

Question 2

Define covariance stationarity

Answer 2

1) Expected(yt) = mu 2) V(yt) = sigma^2 < infinity 3) E[(y_t - mu) (y_(t-s) - mu)] depends on s, not on t These are unconditional statistics. Conditionally, they may vary over time

Question 3

Define strict stationarity

Answer 3

Joint distribution of whole process does not depend on time. Var can be infinite, but constant

Question 4

What are the main causes of non-stationarity?

Answer 4

Seasonality

Time trends

Random walks

Structural breaks

Question 5

Define ergodicity

Answer 5

If two samples from the same process drawn far aprt in time are independent, the process is ergodic. It implies that avefages converge to their expectations if they exist

Question 6

Define (univariate) white noise

Answer 6

1) Zero in expectation 2) Constant, finite variance 3) Zero autocovariance Does not have to be standard normal, although it often is Can be dependent, although not linearly, e.g. GARCH

Question 7

When is an ARMA stationary?

Answer 7

For ARMA(0,Q), it is stationary if errors are white noise For ARMA(1,Q), it is stationary if abs(phi) <1, and errors are white noise For ARMA(1,Q), errors must be finite and roots of characteristic polynomial bust be within unit circle

Question 8

What is the general form of the characteristic equation and under which condition will the process be stationary

Answer 8

Question 9

What is the characteristic polynomial for an ARMA(2,Q)

Answer 9

z^2 - z*phi_1 - phi_2 = 0

Question 10

In which region must phi be for an ARMA(2,Q) to be stationary (given the error is white-noise)?

Answer 10

In the (phi1, phi2) space, within the triangular region bounded by: (-2,-1) (2,-1) and (0,1)

Question 11

What is the autocorrelation function (ACF)

Answer 11

ACF(s) is the s’th autocovariance divided by the variance

Question 12

What is the partial autocorrelation function?

Answer 12

PACF(s) is the s’th slope coefficient in an AR(s) regression. It is the effect of the s’th lag controlling for shorter lags

Question 13

How can you do inference on the ACF?

Answer 13

Simple t-test for individual ACF. Testting for multiple ACFs can be done with the Ljung-Box Q statistic. This is however not heteroscedasticity robust, so it may be advisable to use the LM test instead

Question 14

What is the Box-Jenkins methodology?

Answer 14

An approach to time-series model selection 1) Identification. Analyze ACF and PACF to identify appropriate model 2) Estimation and diagnostics

Question 15

What are some important considerations for model diagnostics?

Answer 15

Are residuals white noise? - Residual plot - Ljung-Box Q stat or Lm test - SACF and SPACF plots Outliers - Visual inspection

Question 16

What is the Ljung-box test?

Answer 16

A test of multiple ACFs. If data is heteroscedastic (most financial data is), it might be advisable to use an LM test, since the Ljung Box Q stat is not robust to heteroscedastcity.

Question 17

What are two key tests to evaluate forecasts?

Answer 17

1) Mincer Zarnowitz regression 2) Diebold-Mariano

Question 18

What is an Mincer-Zarnowitz test?

Answer 18

Regression realized values of forecasts. Null: alpha=0, beta=1 Use Wald, LM or LR Can be generalized to also regress on other variables in the information set. Tests if forecast errors are unforecastable

Question 19

What is the Diebold-Mariano test?

Answer 19

Tests relative performance of 2 forecasts. Makes inference on difference in loss function, typically MSE. Since errors may have serial correlation, must estimate variance with Newey-West estimator Test-statistic is distributed N(0,1)

Question 20

What is the difference between a unit root and a RW?

Answer 20

Unit root processes are generalizations of the simple random walk.

Question 21

What are some problems with unit roots

Answer 21

Exploding variance Inconsistent parameter estimates Suprious regression No mean-reversion

Question 22

What is a Dickey-Fuller test?

Answer 22

It is a test for unit root. In the simplest version, LHS is difference, RHS is level times parameter + error. The null is that the parameter is zero. This can be simply estimated, but inference is harder. Test-statistics follow Dickey-fuller distribution. Non-standard distribution, since variance explodes under the null Time-trends can be included in the regression. Doing so when they are irrelevant decreases power, but failing to do so gives a test with no power. Power can also sometimes be included if more lagged differences are included (AFD) - this models short-run dynamics around the random walk-component

Question 23

What is a Dickey-Fuller test?

Answer 23

It is a test for unit root. In the simplest version, LHS is difference, RHS is level times parameter + error. The null is that the parameter is zero. This can be simply estimated, but inference is harder. Test-statistics follow Dickey-fuller distribution. Non-standard distribution, since variance explodes under the null. Dickey-fuller tests generally have low power Time-trends can be included in the regression. Doing so when they are irrelevant decreases power, but failing to do so gives a test with no power. Power can also sometimes be included if more lagged differences are included (AFD) - this models short-run dynamics around the random walk-component

Question 24

What is a Markov-Switching model?

Answer 24

A model that randomly switches between two regimes, with difference means. Uses transition matrix with 4 probabilities: probability of high and low state given being in either high or low state.

Question 25

What do the ACF and PACF looks like for white noise?

Answer 25

Question 26

What do the ACF and PACF look like for an AR 1 with phi between 0 and 1

Answer 26

Question 27

What do the ACF and PACF look like for an AR(1) with 0>phi>-1

Answer 27

Question 28

What do the ACF and PACF look like for MA(1) with theta between 0 and 1

Answer 28

Question 29

What do the ACF and PACF look like for MA(1) with 0>theta>-1

Answer 29

Question 30

How to derive the equation for ADF?

Answer 30

Start with AR(P) on levels. Add and subtract to get differences until only y_(t-1) is found in levels and all other variables are differences