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Autoregressive (AR) Models

Autoregressive (AR) models are fundamental to time series analysis. They are estimated via regressing a variable on one or more of its lagged values. That is, AR models take the form: \(Y_t = c + \sum_{i = 1}^{p} \beta_i Y_{t-i} + \epsilon_t\) Where we say p is the order of our auto regression. Their estimation in statistical software packages is generally straightforward.

For additional information, see Wikipedia: Autoregressive model.

Keep In Mind

  • An AR model can be univariate (scalar) or multivariate (vector). This may be important to implementing an AR model in your statisical package of choice.
  • Data should be properly formatted before estimation. If not, non-time series objects (e.g., a date column) may be interpereted by software as a time series variable, leading to erroneous output.

Implementations

Following the instructions for creating and formatting Time Series Data, we will use quaterly GDP data downloaded from FRED as an example.

R

#load data
gdp = read.csv("https://github.com/LOST-STATS/lost-stats.github.io/raw/source/Time_Series/Data/GDPC1.csv")

#estimation via ols: pay attention to the selection of the 'GDPC1' column. 
#if the column is not specified, the function call also interprets the date column as a time series variable!
ar_gdp = ar.ols(gdp$GDPC1)
ar_gdp

#lag order is automatically selected by minimizing AIC 
#disable this feature with the optional command 'aic = F'. Note: you will also likely wish to specify the argument 'order.max'.
#ar.ols() defaults to demeaning the data automatically. Also consider taking logs and first differencing for statistically meaningful results.

STATA

*load data
import delimited "https://github.com/LOST-STATS/lost-stats.github.io/raw/source/Time_Series/Data/GDPC1.csv", clear

*Generate the new date variable
*To generalize to a different set of data, replace '1947q1' with your own series' start date.
generate date_index = tq(1947q1) + _n-1

*Index the new variable format as quarter
format date_index %tq

*Convert a variable into time-series data
tsset date_index

*Specifiy and Run AR regression: this STATA method will not automatically select a lag order.
*The 'L.' operator indicates the lagged value of a variable in STATA, 'L2.' its second lag, and so on.
reg gdpc1 L.gdpc1 L2.gdpc1
*variables are not demeaned automatically by STATA. Also consider taking logs and first differencing for statistically meaningful results.