Time Series Analysis Flashcards
A time series
set of observations for a variable over time
time series model
captures the time-series pattern and allows us to make predictions about the variable in the future.
primary limitation of trend models
not useful if the residuals exhibit serial correlation.
autoregressive model (AR)
dependent variable is regressed against one or more lagged values of itself
dependent variable is regressed against one or more lagged values of itself
time series being modeled is covariance stationary
A time series is covariance stationary if it satisfies the following three conditions:
1- Constant and finite expected value.
2- Constant and finite variance.
3- Constant and finite covariance
one-period-ahead forecast for an AR(1) model
xt+1=b0+b1xt
two-step-ahead forecast for an AR(1) model
xt+2=b0+b1xt+1
When an AR model is correctly specified, the residual terms
will not exhibit serial correlation
Serial correlation (or autocorrelation) means the error terms
positively or negatively correlated.
When the error terms are correlated
standard errors are unreliable and t-tests can incorrectly show statistical significance or insignificance.
mean reversion
tendency to move toward its mean
mean reversion formula
xt=b0(1−b1)
root mean squared error(RMSE)
accuracy of autoregressive models in forecasting out-of-sample values
lower RMSE for the out-of-sample data
lower forecast error and will be expected to have better predictive power in the future.
The procedure to test whether an AR time-series model is correctly specified involves three steps:
1-Estimate the AR model being evaluated using linear regression.
2-Calculate the autocorrelations of the model’s residuals.
3-Test whether the autocorrelations are significant.
Random Walk
the predicted value equals the value of the series in the previous period plus a random error term.
random walk equation
xt = xt–1 + εt
Random Walk with a Drift Equation
xt = b0 + b1xt–1 + εt
Random Walk with a Drift
the intercept term is not equal to zero.
Neither a random walk nor a random walk with a drift exhibits
covariance stationarity
a random walk, with or without a drift, exhibits
unit root (b1 = 1).
A time series has a unit root if
the coefficient on the lagged dependent variable is equal to one.
To determine whether a time series is covariance stationary
(1) run an AR model and examine autocorrelations, or
(2) perform the Dickey Fuller test.
first differencing
If we believe a time series is a random walk (i.e., has a unit root), we can transform the data to a covariance stationary using first differencing
Seasonality in a time series is tested by
calculating the autocorrelations of error terms.
autoregressive conditional heteroskedasticity (ARCH)
problem associated with the correlation of variances of the error terms
autoregressive conditional heteroskedasticity (ARCH) exists if
the variance of the residuals in one period is dependent on the variance of the residuals in a previous period
he ARCH(1) regression model is expressed as:
ε2t =a0+a1ˆε2t−1+μt
If a1, is statistically different from zero, the time series is ARCH(1).
ARCH model can be used to predict the variance of the residuals in future periods
σ2t+1=a0+a1ε2t
Cointegration
means that two time series are economically linked or follow the same trend and that relationship is not expected to change
To test whether two time series are cointegrated
(1) if neither time series has a unit root, then the regression can be used;
(2) if only one series has a unit root, the regression results will be invalid;
(3) if both time series have a unit root and are cointegrated, then the regression can be used;
(4) if both time series have a unit root but are not cointegrated, the regression results will be invalid.
to determine whether two times series are cointegrated.
Dickey Fuller test with critical t-values is used
RMSE equals
the square root of the average squared error.
