forecastit 2. linear regression & model statistics

Copyright 2010 DeepThought, Inc. 1

Linear Regression and Model Statistics

Lesson #2

Linear Regression Method



Method Introduction• One of the simpler methods to use for forecasting• Estimates a line through the data• Uses the estimated line equation to forecast future values.• Method format:

– Y = a + b × t



Model Characteristics• Method characteristics

– Fits a line to the data– Estimating a line which minimizes the errors between actual

data points and model estimates• When to use method

– Estimate trend– Estimate trend magnitude

• When not to use– Estimate anything beyond a simple linear relationship



Forecasting Steps1. Set an objective2. Build model3. Evaluate model4. Use model



Objective Setting• Simpler is better• Linear regression allows to test whether a line fitted to the data

works as a model. Objectives should take that principal under consideration

• Example objectives for M2 Money Stock (see next slide):– Test if M2 has a linear trend over time– If M2 exhibits a statistically significant trend, review and

interpret results– If model looks good, create a forecast based off model



Example: M2 Money Stock

May-79 Nov-84 May-90 Oct-95 Apr-01 Oct-06 Apr-120.0

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M2 Money Stock (Billions of $'s)



Method Selection• Observe time series qualities: trend, seasonality, cyclicality, and

randomness• Adjust time frame, units, periods to forecast as needed• Determine if linear regression is a possible candidate based on

method characteristics– Determine if transforming the units will enable use of model

▪ Eight different unit transformation techniques



Build Model• Software finds us the best fit line to the data; minimizing the sum of

squared errors


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Evaluate Model• Descriptive Statistics

– Mean– Variance & Standard Deviation

• Accuracy / Error– SSE– RMSE– MAPE– R2; Adjusted R2

• Statistical Significance– F-Test– P-Value F-Test



Descriptive StatisticsMean

• The average value of the data set

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Variance & Standard Deviation• The sum of squared deviations of the data from the mean

– Estimates the variation the data exhibits from the mean• Standard deviation is the squared root of the variance

– Used to measure the distribution of the variable away from the mean, most observations of the variable will be within ± 3 standard deviations



M2 Example• Mean

– 4214.38

• Variance– 3346475.10

• Standard Deviation– 1829.34


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Accuracy/ErrorSSE

• Sum of Square Errors (SSE)– Sums the errors between the actual values and model values

• Measures the total error of the model• M2 Example:

– SSE: 316778645.89


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RMSE

• The square root of the sum of square error divided by the number of observations

• An averaged out total of errors based upon the number of observations

• Simple way to compare models based on error• M2 Example:

– RMSE: 456.82



MAPE

• The average percentage error of the model• Describes the average percentage of variation exhibited between

actual and forecasted values• M2 Example:

– MAPE: 10.09%


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R-Squared & Adjusted R-Squared

• A proportion between unexplained and explained errors• Measures the percentage of variation captured by the model• Adjusted R2 incorporated the number of variables used and sample

size to adjust the R2 value



M2 Example• R2

– 93.76%

• Adjusted R2

– 93.76%


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Statistical SignificanceF-Test

• A proportion between explained and unexplained errors of model• Used to test if model build is statistically significant from being

equal to zero• The larger the F-test the better



F-Test P-Value

• The F-Test P-Value representsthe percentage of significance of the F-test (blue area on graph)

• The higher the value of the F-test the lower the shaded blue area is. As the blue area decreases, confidence about our model being statistically significant increases

• 1 – p-value = Significance Level of the Model (%)• Significance level of the model (%) represents the amount of

confidence we have that our model is different from a model with no impact, or zero impact



M2 Example


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M2 Money Stock (Billions of $'s)• F-Test– 22778.98

• F-Test P-Value– 0.00



Compare Multiple Models• Skip this step until have knowledge of multiple methods• Will use accuracy/error statistics to compare multiple models to

find best models



Use Model

• Understand limitations of model– Only measures a trend– A long term average

• Answer objectives– Does M2 has a linear trend– If trend exists, what is its magnitude– If model statistically significant, forecast



M2 Example• M2 = 1145.31 + 4.04 × Time• Next Period is 1519• Forecast for that period is:

– Y = 1145.31 + 4.04 × 1519Y = 7283.446866

forecastit 2. linear regression & model statistics

Economy & Finance