What is this all about?

Suppose given a univariate time series data \(x_t\), \(t=1,\ldots,T\), where \(t\) refers to time and \(T\) to the sample size. We would like to have a statistical/probability model \(\{X_t\}_{t\in \mathbb{Z}}\) for the series, where \(X_t\)’s are random variables. Having a statistical model, we can use it for forecasting (prediction), which is one of the basic tasks of time series analysis.

Note: We shall focus on the situations where there is some dependence across times \(t\). Typically, there should be some “physical” reason for temporal dependence. Examples?

This lecture is about a classical approach to time series modeling based on the decomposition (possibly after a preliminary transformation): \[ X_t = T_t + S_t + Y_t,\quad t\in \mathbb{Z}, \] where \(\{T_t\}_{t\in\mathbb{Z}}\) is a (typically deterministic) model for trend, \(\{S_t\}_{t\in\mathbb{Z}}\) is a (typically deterministic periodic) model for seasonal variations, and \(\{Y_t\}_{t\in\mathbb{Z}}\) is a stationary time series model.

header 2

A first code block:

    cd "C:\Users\micha\OneDrive\Documents\Research\TimeSeries\do-files" 
    
    use ../data/NelsonPlosserData.dta, replace
    
    keep l* year bnd
    
    foreach v of var bnd-lsp500{
        gen time_`v' = .
        replace time_`v' = year if `v' ~= .
        quietly summarize time_`v'
        replace time_`v' = time_`v' - r(min)
        }
(111 missing values generated)
(71 real changes made)
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header 3

A second, later code block:

    *   Table & Test for ONE Variable
    
    local x = "lrgnp"
    local k = 1
    
    *   Dickey-Fuller
    
    dfuller `x', trend reg lags(`k')
running C:\Users\micha\OneDrive\Documents\Research\TimeSeries\rmd-files\prof> .do ...





Augmented Dickey-Fuller test for unit root         Number of obs   =        60

                               ---------- Interpolated Dickey-Fuller ---------
                  Test         1% Critical       5% Critical      10% Critical
               Statistic           Value             Value             Value
------------------------------------------------------------------------------
 Z(t)             -2.994            -4.128            -3.490            -3.174
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.1338

------------------------------------------------------------------------------
D.lrgnp      |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lrgnp |
         L1. |  -.1753423   .0585665    -2.99   0.004     -.292665   -.0580196
         LD. |   .4188873   .1209448     3.46   0.001     .1766058    .6611688
      _trend |   .0056465   .0018615     3.03   0.004     .0019174    .0093757
       _cons |   .8134145   .2679886     3.04   0.004     .2765688     1.35026
------------------------------------------------------------------------------