Amount of Text Messaging



Amount of Text Messaging

Sometimes, a nonlinear relationship between the dependent and independent variables is more appropriate than a linear relationship. In such cases, running a linear regression will not be optimal. If the linear model is not the correct form, then the slope and intercept estimates and the fitted values from the linear regression will be biased, and the fitted slope and intercept estimates will not be meaningful (Hox, 2010). Over a restricted range of independent or dependent variables, nonlinear models may be well approximated by linear models (this is in fact the basis of linear interpolation), but for accurate prediction a model appropriate to the data should be selected. A nonlinear transformation should first be applied to the data before running a regression.

Diagnostic Results

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Heteroskedasticity

Micronumerosity

Outliers

Nonlinearity

W-Test

Hypothesis Test

Approximation

Natural

Natural

Number of

Nonlinear Test

Hypothesis Test

Variable

p-value

result

result

Lower Bound

Upper Bound

Potential Outliers

p-value

result

Y

no Problems

-7.86

671.70

2

Variable X1

0.2543

Homoskedastic

no problems

-21377.95

64713.03

3

0.2458

linear

Variable X2

0.3371

Homoskedastic

no problems

77.47

445.93

2

0.0335

nonlinear

Variable X3

0.3649

Homoskedastic

no problems

-5.77

15.69

3

0.0305

nonlinear

Variable X4

0.3066

Homoskedastic

no problems

-295.96

628.21

4

0.9298

linear

Variable X5

0.2495

Homoskedastic

no problems

3.35

9.38

3

0.2727

linear

Statistical Summary

Sometimes, certain types of time-series data cannot be modeled using any other methods except for a stochastic process, because the underlying events are stochastic in nature. For instance, you cannot adequately model and forecast stock prices, interest rates, price of oil, and other commodity prices using a simple regression model, because these variables are highly uncertain and volatile, and does not follow a predefined static rule of behavior, in other words, the process is not stationary (Snijders, 2011). Stationary is checked here using the Runs Test while another visual clue is found in the Autocorrelation report (the ACF tends to decay slowly). A stochastic process is a sequence of events or paths generated by probabilistic laws. That is, random events can occur over time but are governed by specific statistical and probabilistic rules. The main stochastic processes include Random Walk or Brownian motion, Mean-Reversion, and Jump-Diffusion. These processes can be used to forecast a multitude of variables that seemingly follow random trends but restricted by probabilistic laws. The process-generating equation is known in advance but the actual results generated are unknown.

Distributive Lags

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P-Values of Distributive Lag Periods of Each Independent Variable

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable

1

2

3

4

5

6

7

8

9

10

11

12

X1

0.8467

0.2045

0.3336

0.9105

0.9757

0.1020

0.9205

0.1267

0.5431

0.9110

0.7495

0.4016

X2

0.6077

0.9900

0.8422

0.2851

0.0638

0.0032

0.8007

0.1551

0.4823

0.1126

0.0519

0.4383

X3

0.7394

0.2396

0.2741

0.8372

0.9808

0.0464

0.8355

0.0545

0.6828

0.7354

0.5093

0.3500

X4

0.0061

0.6739

0.7932

0.7719

0.6748

0.8627

0.5586

0.9046

0.5726

0.6304

0.4812

0.5707

X5

0.1591

0.2032

0.4123

0.5599

0.6416

0.3447

0.9190

0.9740

0.5185

0.2856

0.1489

0.7794

 

 

 

 

 

 

 

 

 

 

 

 

Periodic

Drift Rate

-1.48%

 

Reversion Rate

283.89%

Jump Rate

20.41%

Volatility

88.84%

 

Long-Term Value

327.72

Jump Size

237.89

 

 

 

 

 

Probability of stochastic model fit:

46.48%

 

 

A high fit means a stochastic model is better than conventional models.

Runs

20

Standard Normal

-1.7321

Positive

25

P-Value (1-tail)

0.0416

Negative

25

P-Value (2-tail)

0.0833

Expected Run

26

 

A low p-value (below 0.10, 0.05, 0.01) means that the sequence is not random and hence suffers from stationarity problems, and an ARIMA

model might be more appropriate. Conversely, higher p-values indicate randomness and stochastic process models might be appropriate.

References

Hox, J. ...