Linear trend model If the variable of interest is a time series, then naturally it is important to identify and fit any trend line deviation time patterns which may be present. Consider again the variable X1 that was analyzed on the page for the mean modeland suppose that it is a time series.
Its graph looks like this: The file containing this data and the models below can be found here. There is indeed a suggestion of a time pattern, namely that the local mean value appears somewhat higher at the end of the series than at the beginning.
There are several ways in which a change in the mean over time could be modeled.
In fact, the sample mean of the first 15 values of X1 is If there is independent evidence for trend line deviation sudden change in the mean in the middle of the sample, then it might make sense to break trend line deviation the data into subsets or else fit a regression model with a dummy variable whose value is equal to zero up the point at which the change trend line deviation and equal to 1 afterward. The estimated coefficient of such a variable would measure the magnitude of the change.
Another possibility is that the local mean is increasing gradually over time, i. If that is the case, then it might be appropriate to fit a sloping line rather than a horizontal line to the entire series.
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This is a linear trend model, also known as a trend-line model. It is a special case of a simple regression model in which the independent variable is just a time index variable, i.
When it is estimated by regression, the trend line is the unique line that minimizes the sum of squared deviations from the data, measured in the vertical direction. More information about this and other properties of regression models is provided in the regression pages on this web site.
If you are plotting the data in Excel, you can just right-click on the graph and select "Add Trendline" affiliate program of binary options with a website the pop-up menu to slap a trend line on it.
Trend and Forecasting
You can also use the trendline options to display R-squared and the estimated slope and intercept, but no other numerical output, as shown here: The intercept of the trend line the point at which the line crosses the y-axis is More trend line deviation can be obtained by fitting the regression model using statistical software such as RegressIt. R-squared for this model is 0. Adjusted R-squared, which is 0.
See this page for a more thorough discussion of R-squared and adjusted R-squared.
So, trend line deviation linear trend model does improve a bit on the mean model for this time series. Is the improvement statistically significant?
To help answer that question, we can look at the t-statistic of the slope coefficient, whose value is 2. These statistics indicate that the estimated slope is different from zero at better than the 0. If the objective of the analysis is to forecast what will happen next, the most important issue in comparing the models is the extent to which they make different predictions.
Or just looking for a way to evaluate the fit of the line to your data or, conversely, the fit of your data to the line? If it's the second one, you can choose the "options" tab in the add trendline dialog, and choose to have excel display the equation of the trendline and the r-squared value.
Which model should be chosen? The data argues in favor of the linear trend model, although consideration should also be given to the question of whether it is logical to assume that this series has a steady upward trend as opposed, say, to no trend or a randomly changing trendbased on everything else that is known about it.
The trend that has been estimated from this sample of data trend line deviation statistically significant but not overwhelmingly so.
Formula: measure deviation from trendline
That means it is very good, right? Well, no. The straight line does not actually do a very good job of capturing the fine detail in the time pattern. This tendency is measured in statistical terms by the lag-1 autocorrelation and Durbin-Watson statistic.
Linear trend estimation
If there is no time pattern, the lag-1 autocorrelation should be very close to zero, and the Durbin-Watson statistic ought to be very close to 2, which is not the case here.
If the model has succeeded in extracting all the "signal" from the data, there should be no pattern at all in the errors: the error in the next period should not be correlated with any previous errors.
The linear trend model obviously fails the autocorrelation test in this case. If we are interested in using the model to predict the future, the fact that 8 out its last 9 errors have been positive and they appear binary option reviews binex be getting worse is cause for concern.
The forecast clearly appears to be too low, given what X2 has been doing lately and given trend line deviation in the past it did not show a tendency to quickly return to the regression line after wandering away from it.
From the above discussion of trends in random data with known variancewe know the distribution of calculated trends to be expected from random trendless data. The use of a linear trend line has been the subject of criticism, leading to a search for alternative approaches to avoid its use in trend line deviation estimation. One of the alternative approaches involves unit root tests and the cointegration technique in econometric studies. The estimated coefficient associated with a linear trend variable such as time is interpreted as a measure of the impact of a number of unknown or known but unmeasurable factors on the dependent variable over one unit of time.
This is significantly less than the standard error of the regression for the linear trend model, which is 2. The random-walk-with-drift model would predict the value of X2 in period 31 to be slightly above its observed value in period 30, which seems more realistic here.
Although trend lines have their uses as visual aids, they are often poor for purposes of forecasting outside the historical range of the data. Most time series that arise in nature and economics trend line deviation not behave as though there are straight lines fixed in space to which they want to return some day.
Rather, their levels and trends undergo evolution. When trying to project an assumed linear trend into the future, we would like to know the current values of the slope and intercept--i. We will see that other forecasting models often do a better job of this than the simple linear trend model. Return to top of page.