Define autocovariance

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

Define covariance stationarity

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

Define strict stationarity

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

What are the main causes of non-stationarity?

Seasonality

Time trends

Random walks

Structural breaks

Define ergodicity

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

Define (univariate) white noise

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

When is an ARMA stationary?

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

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

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

z^2 - z*phi_1 - phi_2 = 0

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

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

What is the autocorrelation function (ACF)

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

What is the partial autocorrelation function?

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

How can you do inference on the ACF?

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

What is the Box-Jenkins methodology?

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

What are some important considerations for model diagnostics?

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

What is the Ljung-box test?

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.

What are two key tests to evaluate forecasts?

1) Mincer Zarnowitz regression 2) Diebold-Mariano

What is an Mincer-Zarnowitz test?

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

What is the Diebold-Mariano test?

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)

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

Unit root processes are generalizations of the simple random walk.

What are some problems with unit roots

Exploding variance Inconsistent parameter estimates Suprious regression No mean-reversion

What is a Dickey-Fuller test?

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

What is a Dickey-Fuller test?

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

What is a Markov-Switching model?

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.

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

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

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

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

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

How to derive the equation for ADF?

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