14 Time series Flashcards
examples of time series data
Yearly GDP of Norway for a period of 20 years
Daily NOK/Euro exchange rate for past year
Quartertly data on the inflation % unemployment rate in the US from 1957-2005
Y_{t-j} is called __
jth lag
jth lag
Y{t-j} is called jth lag
Y_{t+j}
jth future value
the first difference
Yt - Y{t-1}
Y_t - Y_{t-1}
the first difference
time series regression models can be used for
estimating (dynamic) causal effects
forecasting
logarithmic growth
logarithmic grow or log-difference:
△ln(Yt) = ln(Yt) − ln(Y{t−1})
(≈ △Yt / Y{t-1})
autocorrelation
The jth autocorrelation
ρj = Cov(Yt, Y{t − j}) / Var(Yt)
the denominator assumes stationarity of Yt
stationarity
A time series is stationary if it’s probability distribution does not change over time
When the joint distribution of a time series variable and its lagged values does not change over time
forecast error
The difference between the value of the variable that actually occurs and its forecasted value:
Y{T+j} − Y{T+j | T}
NOT the same as residual
A forecast Y{T+j | T} and forecast error for j ≥ 1 are “out-of-sample”: They are calculated for some date beyond the data set used to estimate the regression. Y_{T+j} is not observed in the data set used to estimate the regression.
RMSFE
Root mean squared forecast error
RMSFE = sqrt( E[ (Y{T+1} − Y{T+1 | T})^2 ] )
AR(1)
First-order autoregressive model:
Yt = β0 + β1Y{t−1} + ut
Forecast in next period based on AR(1) model:
Y{T+1 | T} = β0 + β1Y_T
AR(p)
The pth order autoregressive model:
Yt = β0 + β1Yt−1 + β2Yt−2 + … + βpYt−p + ut
The number of lags p is called the order or lag length of the autoregression.
(AR(p))
We can use ___ to determine the lag order p
(AR(p))
We can use t- or F-tests to determine the lag order p
(or using an information criterion: BIC, AIC)
ADL model
Autoregressive distributed lag model
A linear regression model in which the time series variable Yt is expressed as a function of lags of Yt and of another variable, Xt.
The model is denoted ADL(p, q), where p denotes the number of lags of Yt and q denotes the number of lags of Xt.
Granger causality test
A procedure for testing whether current and lagged values of one time series help predict future values of another time series
two types of trends
Deterministic trend
Yt = β0 + λt + u1
(series is a nonrandom function of time)
Stochastic trend
Yt = β0 + Y{t−1} + u1
(series is a random function of time)
random walk
A time series process in which the value of the variable equals its value in the previous period plus an unpredictable error term:
Yt = Y{t-1} + ut
where ut is i.i.d.
random walk with drift
A generalization of the random walk in which the change in the variable has a nonzero mean but is otherwise unpredictable.
Yt = β0 + Y{t-1} + ut
A random walk is nonstationary, as the distribution is not constant over time. The variance of a random walk increases over time:
Var(Yt) = Var(Y{t−1}) + Var(u{t})
If Yt follows and AR(p) model, Yt is stationary if ___
If Yt follows and AR(p) model, Yt is stationary if its roots z are all greater than 1 in absolute value.
The roots are the values of z that satisfy
1 − β{1}z − β{2}z^2 − … − β{p}z^p = 0
method for detecting trends
Dickey-Fuller test for AR(1)
Yt = β0 + β1Y{t−1} + ut
H0 : β1 = 1 vs H1 : β1
unit root
an autoregression with a largest root equal to 1
Instead of testing the null hypothesis of a stochastic trend against the alternative hypothesis of no trend….
…the alternative hypothesis can be that Yt is stationary around a deterministic trend. The Dickey-Fuller regression then includes a deterministic trend
If the series Yt has a stochastic trend, then ___ does not have a stochastic trend.
If the series Yt has a stochastic trend, then the first difference of the series △Yt does not have a stochastic trend.
