# Time Series Analysis: Simple and Log-linear Trend Models

Simple Time Series Models

This is basic trend modeling.

A simple trend model can be expressed as follows:

$\tilde{y}_t = b_0 + b_1 t + \varepsilon_t$
• $b_0$ = the y-intercept; where t = 0.
• $b_1$ = the slope coefficient of the time trend.
• t = the time period.
• $\tilde{y}_t$ = the estimated value for time t based on the model.
• $\varepsilon_t$ = the random error of the time trend.

The big validity pit-fall for simple trend models is serial correlation; if this problem is present, then you will see an artificially high R2 and your slope coefficient may falsely appear to be significant.

There is a visual way to detect serial correlation (not shown) or you can perform a Dubin-Watson test.

Log-linear Trend Models

This applies to non-linear time series trends.

The structure is:

$\ln y_t = b_0 + b_1 t + e_t$$y_t = e^{b_0 + b_1 t + e_t}$

Again, like the simple trend model, use a graph or Durbin Watson test to check for serial correlation, as this will be a big threat to validity.

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