Time Series Flashcards
time series
gives us value of the same variable Y at different time periods
lags
Yt-1, Yt-2 etc
first difference
change in value of Y between time t-1 and t
autocorrelated
when a series is correlated with its lags
volatility clustering
when there are periods of high volatility followed by periods of low volatility
breaks
abrupt or occur slowly due to econ policy or changes in structure of economy
serial correlation (autocorrelation)
correlation between error terms (at time t, t-1, t-2 etc) in regression model
exogeneity assumption
ut must be uncorrelated with all xts i.e. all explanatory variables (X) cannot respond to change in/past values of dependent variable (Y)
no autocorrelation assumption
e.g if interest rate is unexpectedly high in one period it shouldn’t be high in the next period too
consequences of autocorrelation
OLS no longer BLUE
OLS se underestimated - CI too narrow - t ratio too large - p values too small - more likely to incorrectly reject null
testing autocorrelation
do regression of residuals et on their lagged values et-1
HAC
Heteroscedasticity and Autocorrelation Consistent standard errors (they take autocorrelation into account)
conditional heteroscedasticity
variance of the error term is autocorrelated (ie.e when it’s high in one period it’s high in the next
arises when dependent variable has volatility clustering
AR(p) model
uses Yt-1 to forecast Yt. p = no. of lags
AR(p) model assumptions
conditional expectation of ut = 0 given past values of Yt
errors are serially uncorrelated
ADL(p,q) model
autoregressive distributed lag model
lagged values of dependent variable are included as regressors
p=lags of Yt, q=lags of additional predictor Xt
least squares assumptions for ADL
error term has conditional mean 0 given all the lags of regressors
random variables have a stationary distribution
no large outliers
no multicollinearity
stationarity
series Yt is stationary if its probability distribution doesn’t change over time
types of non-stationary
trends and breaks
leads to bias an inconsistency
deterministic trend
variable is a linear function of time (would indicate that growth rate is constant over time)
stochastic trend
trend is random and varies over time (UK GDP growth rate is not constant)
random walk
Yt = Yt-1 + ut
if Yt follows an AR(1) with B1=1 then Yt contains a stochastic trend and is non-stationary
random walk with drift
Yt = B0 + Yt-1 + ut
problems with stochastic trends
biased coeff estimates
non-normal distributions of t-stat
spurious regressions
testing for a unit root
Dickey-Fuller test
Dickey-Fuller test
regressing Yt on its lag
testing for the existence of a stochastic trend in the presence of a deterministic trend (we include an intercept and a time trend into spec on unit root test)
Augmented Dickey-Fuller test
if AR(1) model doesn’t capture all the serial correlation in Yt then DF test is invalid
differencing
eliminates stochastic trend
transforming non-stationary time series to stationary
Chow test
to test fr breaks in ADL and DL models in a subset of parameters only
BIC (lag selection)
Bayes Information Criteria - want to choose p that minimises BIC
AIC (lag selection)
Akaike Information Criteria - want to choose p that minimises AIC
SSR (p)
sum of squared residuals of a model estimated with p lags
focus of forecasting
how good is a model at predicting future events (not to estimate causal effects)
to evaluate forecasting model (and compare different models)
adjusted R^2
RMSFE
out-of-sample forecasting performance
RMSFE
the root mean squared forecast error = size of typical mistake we make when using forecasting model (smaller means better model)
adjusted R^2
how well does the model explain the variation in the dependent variable (higher means explains more of variation)
out-of sample forecasting performance
how well is the model performing in real time
forecast errors
mistake made when forecasting (not the same as predicted residuals)
pseudo out-of-sample forecasting
method for simulating real time performance of a fc model using historical data for the series Y up to period T
standard deviation of pseudo out-of-sample forecast errors provide
estimate of RMSFE
95% forecast interval
interval that contains the future value of the series 95% of the time
Granger causality tests
tests of the predictive content of the predictors in a forecasting model
stat = F-stat
the predictor “Granger causes” w/e
dynamic causal effect
follow time path of the effect of a shock over time e.g. effect of increasing IR on US GDP
DL Model
distributed lag model - used to estimate dynamic effect
DL model assumptions
exogeneity
random variables have a stationary distribution
no large outliers
no multicollinearity
exogeneity
condition that guarantees that the estimated coeffs can be interpreted as causal effects
- will not hold if there are omitted variables in error term that are correlated with past or present values
- holds if lags don’t effect w/e beyond last lag
