One of the plot represent the linear trend added to white noise $y_t=\beta_0+\beta_1t+w_t$, and $w_t=$wn$(0,\sigma^2)$.

Other one is random walk with constant drift $y_t=\delta +y_{t-1}+\epsilon_t$ and $\epsilon_t=$wn$(0,\tau^2)$.

enter image description here

Which plot is for which model. Guess the parameters for the model?

I tried plot.ts(0.5+0.6*(1:100)+rnorm(100)) which is looking more like left plot hence it should be linear trend added to white noise.

How to guess the parameters just from the plot?

  • 2
    $\begingroup$ Hint: what is the autocorrelation of the residuals in each model? $\endgroup$
    – whuber
    Nov 20 '19 at 19:41

yt=δ+yt−1+ϵt is the leftmost plot while yt=β0+β1t+ϵt is the rightmost plot. Both original time series have very similar acf's and pacf's thus model identification is difficult as the two competing models can't be distinguished from each other simply using the observed data.

However if one incorrectly specifies yt=β0+β1t+ϵt for the leftmost series the residuals will not be free of structure suggesting misidentification. Similarly if you specify yt=δ+yt−1+ϵt for the rightmost data set you will also get a residual series that is not free of structure.

Discerning between a stochastic/adaptive model and a fixed/deterministic model can get a little bit trickier when you have break points in the data and possible level/step shifts with possible error variance or model parameter transience.

Model identification strategies need to consider the potential of a hybrid model where both stochastic and/or deterministic structure are possibly present in the data as described here in an iterative process https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf.

Ultimately it is the analysis of model residuals that is the final arbiter of which approach is more correct for any given data set.

  • $\begingroup$ Your initial sentence asserts both plots use identical models--obviously a typo somewhere. Maybe it's time to learn how to use the MathJax markup? It's easy to do. I also urge you to rethink your assertion about "very similar acfs," because the residuals from a regression of the two plots will have strikingly different acfs. $\endgroup$
    – whuber
    Nov 21 '19 at 15:23
  • $\begingroup$ my reflection about "similar acf/pacf" was for the original series as i stated time series ,,,not for residuals from any model . The residuals from a model are never (for me ) referred to as a time series without the adjective "residua" $\endgroup$
    – IrishStat
    Nov 21 '19 at 15:27
  • $\begingroup$ Unfortunately, the entire point of the question is to help the student learn to use and think about residuals. Your suggestion that the acf/pacf of the data as given would be helpful and appropriate is a misleading one, IMHO. Regardless, if you want your answer even to make sense then you need to fix the error at the outset. $\endgroup$
    – whuber
    Nov 21 '19 at 15:32
  • $\begingroup$ @whuber and IrishStat I understand that I need to look at residuals. But if just given a ts how can I know whether it is linear trend with noise or random walk with drift without peeking at residuals? This was one of the questions in an exam $\endgroup$
    – Vineet
    Nov 27 '19 at 14:08
  • $\begingroup$ just from a visual the rightmost plot is more FIXED as values folow the trend tightly and future values are somwhat more pre-deternined with little or no adaptation ...where the leftmost plot suggests that each point is primarily based upon the previous point such that there is more wiggle thus adaptive in nature or auto-regressive in nature as older values seem to be less informative... $\endgroup$
    – IrishStat
    Nov 27 '19 at 14:49

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