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It is said that the time series has a stochastic trend if the first autocorrelation coefficient will be near 1.

Q1) What does it mean by the above statement?

Q2) How do we calculate the first autocorrelation of a time series?

Any comments on the matter will be well-appreciated.

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3 Answers 3

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you hypothesize that your data is generated by the following process: $y_t=\phi_0+\sum_{k=1}^P\phi_ky_{t-k}+\varepsilon_t$, where $\phi_k$ are your autocorrelation coefficients, and $\varepsilon_T$ - random errors. Next, you estimate your $\phi_t$ using one of the methods of estimation of autoregressive processes AR(P) of order P, e.g. see AR(P), there's no point in doing manually, use statistical packages like Stata.

you have to choose order P, there are ways of doing it, such as trying many different Ps and comparing their AIC

next, you test your model for the unit root. in case of P=1, i.e. $y_t=\phi_0+\phi_1y_{t-1}+\varepsilon_t$, this is testing whether $\phi_1=1$. there are ways of testing this hypothesis too. if it's very close to 1 then you have a unit root.

what does this mean? it means that your $y_t$ will drift away over time, it's non-stationary. it is not a deterministic trend like $y_t=\beta t$, where you know where and how fast it's going. on the other hand you know that $|E[y_\infty]|=\infty$. if you have no unit root then the process is stationary and in the long run your $y_\infty$ will not deviate too much from where it is now.

examples. interest rates are stationary, they don't have stochastic trends. in ancient Egypt some 4-5 thousands years ago the interest rate on the unsecured personal loans were around 20% annual.

asset prices are not stationary, they have drifts, possible both stochastic and deterministic. look at house prices 50 years ago and now, they grew maybe 10-folds if not more.

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Autocorrelation is the correlation of a series with itself. Suppose $X = {X_1, X_2, X_3, ...}$ is your time series. Then the autocorrelation between $X_t$ amd $X_s$ is:

$$ \frac{E[(X_t-\mu_t)(X_s-\mu_s)]}{\sigma_t \sigma_s} $$

This can be simplified quite a lot if the series you have is stationary (a common assumption), in which case the autocorrelation depends only on the lag $k$:

$$ \frac{E[(X_t-\mu)(X_{t+k}-\mu)]}{\sigma^2} $$

In your case, (for Q2), the lag $k$ is 1. In practice, this means if you have a time series of $x_1, x_2, ..., x_n$ elements, you would calculate the correlation (using Excel CORR() perhaps) between the sub-series $x_1, x_2, ..., x_{n-1}$ and $x_2, x_3, ..., x_n$.

A positive (negative) autocorrelation means that an increase in your time series is often followed by another increase (a decrease). If the autocorrelation is close to 1, then an increase is almost certainly followed by another increase. In other words, the average value of the time series is increasing. Alternatively, a decrease is almost certainly followed by a decrease. In other words, the average level of the time series is decreasing. The "trend" part follows trivially.

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1.) Autocorrelation is the correlation of a time series against the lagged version of itself.

2). First autocorrelation is the correlation of the time series against the lag(1) version of itself.

Let's look at the example below

Period_Numbers = [1,2,3,4,5,6,7,8,9,10]
Time_Series = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]

First Autocorrelation is calculated by taking the correlation of periods 1 to 9 and periods 2 to 10. That is

Autocorrelation_lag1 = Correlation of [(10, 20, 30, 40, 50, 60, 70, 80, 90), (20, 30, 40, 50, 60, 70, 80, 90, 100)]

Here is a an explanation from Investopedia - http://www.investopedia.com/terms/a/autocorrelation.asp

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