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13. Several Aspects of Regression Analysis
This chapter focusses on topics that add to the understanding of regression analysis. This includes alternative specifications for these models, and what happens in the situations where basic regression assumptions are violated.
13.1. Developing models
The goal when developing a model is to approximate the complex reality as close as possible with a relatively simple model, which can then be used to provide insight into reality. It is impossible to represent all of the influences in the real situation in a model, instead only the most influential variables are selected.
Building a statistical model has 4 stages:
- Model Specification: This step involves the selection of the variables (dependent and independent), the algebraic form of the model, and the required data. In order to this correctly it is important to understand the underlying theory and context for the model. This stage may require serious study and analysis. This step is crucial to the integrity of the model.
- Coefficient Estimation: This step involves using the available data to estimate the coefficients and/or parameters in the model. The desired values are dependent on the objective of the model. Roughly there are two goals:
- Predicting the mean of the dependent variable: In this case it is desirable to have a small standard error of the estimate, se. The correlations between independent variables need to be steady, and there needs to be a wide spread for these independent variables (as this means that the prediction variance is small).
- Estimating one or more coefficients: In this case a number of problems arise, as there is always a trade-off between estimator bias and variance, within which a proper balance must be found. Including an independent variable that is highly correlated with other independent variables decreases bias but increases variance. Excluding the variable decreases variance, but increases bias. This is the case because both these correlations and the spread of the independent variables influence the standard deviation of the slope coefficients, sb.
- Model Verification: This step involves checking whether the model is still accurate in its portrayal of reality. This is important because simplifications and assumptions are often made while constructing the model, this can lead to the model becoming (too) inaccurate). It is important to examine the regression assumptions, the model specification, and the selected data. If something is wrong here, we return to step 1.
- Interpretation and Inference: This step involves drawing conclusions from the outcomes of the model. Here it is important to remain critical. Inferences drawn from these outcomes can only be accurate if the previous 3 steps have been completed properly. If these outcomes differ from expectations or previous findings you must be critical about whether this is due to the model or whether you really have found something new.
13.2. Further Application of Dummy Variables
Dummy variables were introduced in chapter 12 as a way to include categorical variables in regression analysis. Further uses for these variables will be discussed here.
Dummy variables have values of either 1 or 0, to represent two categories. It is also possible to represent more than two categories by using a combination of multiple dummy variables. The rule is: number of categories -1 = number of dummy variables. So for three categories, two dummy variables are used. For example:
Yes: x1= 1 x2=1
Maybe: x1=1 x2=0
No: x1=0 x2=0
Time series data can be portrayed by dummy variables in the same manner. In this case time periods are the categories.
Dummy variables are becoming more popular as a tool in experimental designs. Here again similar specification is used to represent several levels of the treatment. In experimental designs there are also so-called blocking variables, which are part of the environment and cannot be randomized or preselected. By using dummy variables these can be included in the model in such a manner that it’s variability can be removed from the independent variables.
13.3. Values in Time-Series Data
When measurements are taken over time the values of the dependent variable are referred to as lagged. Such time-series observations are specified in formulates with the subscript “t”.
It is important to be aware of lagged values because the value of the dependent variable in one time period is often related to the value of a previous time period. This previous value of the dependent variable is called a lagged dependent variable.
With lagged dependent variables there is no difference in coefficient estimation, confidence intervals and hypothesis tests in comparison to a regular dependent variable. It is, however, advised to be cautious with the use of confidence intervals and hypothesis tests, as it is possible that the equation errors, εi, are no longer independent, which leads to the coefficient estimates no longer being efficient (though unbiased), which means that confidence intervals and hypothesis tests are no longer valid. If the equation errors remain independent, the quality of the approximation will improve as the number of sample observations increases.
13.4. Inclusion of proper independent variables
Models can never be as complete as reality because a model cannot contain al variables that are likely to affect the dependent variable in the real world, so a selection must be made. The joint influence of the variables that are not selected are then absorbed in the error term. If, however, an important variable is omitted this means that the estimated coefficients of the other independent variables will be different, and any conclusions drawn from this model may be faulty.
13.5. Multicollinearity of independent variables
It is possible for two independent variables to be highly correlated. In this case the estimated coefficients of the model can be very misleading. This phenomenon is referred to as multicollinearity. This problem arises out of the data itself, and sadly there is little that can be done about it. It is still important to be watchful of this though. Indications for multicollinearity are:
- The regression coefficients are very different from previous research or expectations.
- Coefficients of variables that are believed to have a strong influence, have a small students t-statistic.
- The student’s t-statistics are small for all coefficients, even though there is a large F-statistic (indicating no individual effect but strong effect for the total model).
- There are high correlations between individual independent variables and/or there is a strong linear regression relationship between independent variables.
There are several possibilities to correct multicollinearity:
- Removing one or more of the highly correlated independent variables (this may also have side effects, see 13.4).
- Changing the specification of the model by including a new variable that is a function of the correlated variables.
- Obtaining additional data where the correlation between the independent variables is weaker.
13.6. Variance of error terms
When one or more of the regression assumptions are violated the least squared method will lead to inefficient estimated coefficients and misleading conclusions. One of these assumptions is the assumption of homoscedasticity, that the error terms are uniformly distributed and are not correlated. This assumptions is violated if a model exhibits heteroscedasticity. There are various ways to check for this:
- Relating the error variance to an alternative explanation.
- Making a scatterplot of the residuals versus the independent variables and the predicted values from the regression. A visible relationship (like errors increasing with increasing X-values) is a sign of heteroscedasticity.
- Testing the null hypothesis that the error terms have the same variance, against the alternative hypothesis that their variances depend on the expected values. This procedure can be used when the predicted value of the dependent variable has a linear relationship with the variance of the error term.
It is also possible that there is simply an appearance of heteroscedasticity, for example if logarithmic model is more appropriate but a linear regression model was estimated instead.
13.7. Correlated error terms
The error term represents all variables that influence the dependent variable, outside of the independent variables. In time-series data this term functions differently, as many of these variables may behave similarly over time. This can thus result in a correlation between error terms, also referred to as auto correlated errors. This means that the estimated standard errors for the coefficients are biased, null hypotheses might be falsely rejected, and confidence intervals would be too narrow.
Assuming all errors have the same variance, the structure for autocorrelation, or the first-order autoregressive model of auto correlated behaviour, is:
Where ρ is a correlation coefficient, and ut is a random variable (thus not auto correlated). The coefficient ρ varies from -1 to +1, where a ρ of 0 signifies no autocorrelation, while a -1 or +1 signifies strong autocorrelation.
Autocorrelation can also be found by time plotting the residuals, a jagged plot signifies no autocorrelation.
A more formal test of autocorrelation is the Durbin-Watson test, based on model residuals. It is calculated as follows:
The Durbin-Watson statistic can be written as: d=2(1-r). Where r is the sample estimate of the population correlation between adjacent errors.
If the errors are not auto correlated, r≈0 and d≈2.
A positive correlation is shown as: 0≤d
A negative correlation is shown as: 2
When there are auto correlated errors then the regression procedure needs to be modified to remove its effects. Estimating the coefficients of such a model follows two steps:
- The model is estimated using least squares, which obtains the Durbin-Watson d-statistic. The r, as autocorrelation parameter, can then be calculated.
- Use least squares to estimate a second regression with:
Dependent variable: yt à (yt – ryt-1)
Independent variable: β1x1t à (xi1 – rx1,t-1) - Divide the estimated intercept from this second model by (1-r) to get the correct estimated intercept for the original model.
- Use the output from the second model to carry out hypothesis tests and confidence intervals.
An even more severe problem presents itself when there is a model with lagged dependent variables and auto correlated errors. Here the model also needs to be modified, using a variation on the procedure explained above.
Samenvatting Statistics for Business and Economics
Multiple Regression (12)
12. Multiple Regression
Simple regression (see chapter 11) can predict a dependent variable as a function of a single independent variable. But often there are multiple variables at play. In order to determine the simultaneous effect of multiple independent variables on a dependent variable, multiple regression is used. The least squares principle fit the model.
12.1. The model
As with simple regression, the first step in the model development is model specification, the selection of the model variables and functional form of the model. This is influenced by the model objectives, namely: (1) predicting the dependent variable, and/or (2) estimating the marginal effect of each independent variable. The second objective is hard to achieve, however, in a model with multiple independent variables, because these variables are not only related to the dependent variable but also to each other. This leaves a web of effects that is not easily untangled.
To make multiple regression models more accurate an error term “ε” is added, as a way to recognize that none of the described relationships in the model will hold exactly and there are likely to be variables that affect the dependent variable, but are not included in the model.
12.2. Estimating Coefficients
Multiple regression coefficients are calculated with the least squares procedure. However, again this is more complicated than with simple regression, as the independent variables not only affect the dependent variable but also each other. It is not possible to identify the unique effect of each independent variable on the dependent variable. This means that the higher the correlations between two or more of the independent variables in a model are, the less reliable the estimated regression coefficients are.
There are 5 assumptions to standard multiple regression. The first 4 are the same as are made for simple regression (see chapter 11). The 5th states that it is not possible to find a set of nonzero numbers such that the sum of the coefficients equals 0. This assumption excludes the cases in which there is a linear relationship between a pair of independent variables. In most cases this assumption will not be violated if the model is properly specified.
Whereas in simple regression the least squares procedure finds a line that best represents the set of points in space, multiple regression finds a plane that best represents these points (as each variable is represented with its own dimension).
It is important to be aware of the fact that in a multiple regression it is not possible to know which independent variable predicts which change in the dependent variable. After all, the slope coefficient estimated is affected by the correlations between all independent and dependent variables. This also means that any multiple regression coefficient is dependent on all independent variables in the model. These coefficients are thus referred to as conditional coefficients. This is the case is all multiple regression models unless there are two independent variables with a sample correlation of zero (but this is very unlikely). Because of this.....read more
Several Aspects of Regression Analysis (13)
13. Several Aspects of Regression Analysis
This chapter focusses on topics that add to the understanding of regression analysis. This includes alternative specifications for these models, and what happens in the situations where basic regression assumptions are violated.
13.1. Developing models
The goal when developing a model is to approximate the complex reality as close as possible with a relatively simple model, which can then be used to provide insight into reality. It is impossible to represent all of the influences in the real situation in a model, instead only the most influential variables are selected.
Building a statistical model has 4 stages:
- Model Specification: This step involves the selection of the variables (dependent and independent), the algebraic form of the model, and the required data. In order to this correctly it is important to understand the underlying theory and context for the model. This stage may require serious study and analysis. This step is crucial to the integrity of the model.
- Coefficient Estimation: This step involves using the available data to estimate the coefficients and/or parameters in the model. The desired values are dependent on the objective of the model. Roughly there are two goals:
- Predicting the mean of the dependent variable: In this case it is desirable to have a small standard error of the estimate, se. The correlations between independent variables need to be steady, and there needs to be a wide spread for these independent variables (as this means that the prediction variance is small).
- Estimating one or more coefficients: In this case a number of problems arise, as there is always a trade-off between estimator bias and variance, within which a proper balance must be found. Including an independent variable that is highly correlated with other independent variables decreases bias but increases variance. Excluding the variable decreases variance, but increases bias. This is the case because both these correlations and the spread of the independent variables influence the standard deviation of the slope coefficients, sb.
- Model Verification: This step involves checking whether the model is still accurate in its portrayal of reality. This is important because simplifications and assumptions are often made while constructing the model, this can lead to the model becoming (too) inaccurate). It is important to examine the regression assumptions, the model specification, and the selected data. If something is wrong here, we return to step 1.
- Interpretation and Inference: This step involves drawing conclusions from the outcomes of the model. Here it is important to remain critical. Inferences drawn from these outcomes can only be accurate if the previous 3 steps have been completed properly. If these outcomes differ from expectations or previous findings you must be critical about whether this is due to the model or whether you really have found something new.
13.2. Further Application of Dummy Variables
Dummy variables were introduced in chapter 12 as a way to include categorical variables in regression analysis. Further uses for these variables will be.....read more
Analysis of Variance (15)
15. Analysis of Variance
There are situations and experiments that require processes to be compared at more than two levels. Data from such experiments can be analysed using analysis of variance or ANOVA.
15.1. Comparing Population Means
There are other ways to compare population means than ANOVA, but these are based on the assumption of either paired observations or independent random samples, and can only be used to compare two population means. ANOVA can be used to compare more than two populations, and also uses assessments of variation, which forms a large problem in other methods.
15.2. One-Way ANOVA
The procedure for testing the equality of population means is called a one-way ANOVA. This procedure is based on the assumption that all included populations have a common variance.
The total sum of squares (SST) in this procedure is made up of a within-group sum of squares (SSW) and a between groups sum of squares (SSG): SST = SSW + SSG
This division of the SST forms the basis of the one-way ANOVA, as it expresses the total variability around the mean for the sample observations.
If the null hypothesis is true (all population means are the same) then both SSW and SSG can be used to estimate the common population variance. This is done by dividing by the appropriate number of degrees of freedom.
Because SSW and SSG both provide an unbiased estimate of the common population variance if the null hypothesis is true, a difference between the two values indicates that the null hypothesis is false. The test of the null hypothesis is thus based on the ratio of mean squares:
Where and
. With the assumptions that the population variances are equal and the population distributions are normal.
The closer the ratio is to 1, the less indication there is that the null hypothesis is false.
These results are also summarized in a one-way ANOVA table, which has the following format:
Source of Variation | Sum of Squares | Degrees of Freedom | Mean Squares | F-ratio |
Between groups | SSG | K – 1 | MSG | MSG/MSW |
Within groups | SSW | n – K | MSW | |
Total | SST | n – 1 |
|
|
It is also possible to calculate a minimum significant difference (MSD) between two sample means, as evidence to conclude whether the population means are different. This is done:
With sp being the estimate of variance (),.....read more
Predictions with Time-Series Data (16)
16. Predictions with Time-Series Data
Time series data involves measurements that are ordered over time, in which the sequence of observations is important. Most procedures for data analysis cannot be used for this data, as these procedures are based on the assumption that the errors are independent. Thus, different forms of analysis are needed.
The main goal of analysing time-series data is to make predictions. An important assumption here is that the relations between variables remain constant.
16.1. Time-Series Components
Most time-series have the following four components:
- Trend component: Values grow or decrease steadily over long periods of time.
- Seasonality component: An oscillatory patterns that is specific per season (quarter year) repeats itself.
- Cyclical component: And oscillatory or cyclical pattern that is not related to seasonal behaviour.
- Irregular component: No pattern is regular enough to only exist through these predictable trends; each series of data will also have irregular components (similar to the random error term).
Analysis of time-series data involves constructing a formal model in which most of these components are explicitly or implicitly present, in order to describe the behaviour of the data series. In building this model the series components can either be regarded as being fixed over time, or as steadily evolving over time.
16.2. Moving Averages
Moving averages are the basis for many practical adjustment procedures. It can be used to remove the irregular component or smooth seasonal component:
- Removing the irregular component: This is done by replacing each observation with the average of itself and its neighbours. The theory is that this will decrease the effect of the irregular component on each data point.
- Smoothing the seasonal component: This is done by producing four-period moving averages in such a manner that the seasonal values become one single seasonal moving average. This does mean that the values have shifted in time (in comparison to the original series), but this can be corrected by centring the averages. The specific procedure always depends on the amount of stability the pattern is assumed to have, and whether seasonality is thought to be additive or multiplicative (in the latter case: use logarithms).
If there is an assumption of a stable seasonal pattern a further seasonal-adjustment approach can be used: the seasonal index method. Here the original series is expressed as a percentage of the centred 4-point moving average series.
Additionally moving averages are very suitable for detecting cyclical components and/or trends.
16.3. Predictions using smoothing
There are a various prediction methods, and the choice you make should always depend on the resources, the objectives, and the available data.
Simple exponential smoothing is a more basic prediction method that is appropriate when the series is non-seasonal and has no consistent trends. It predicts future values on the basis of an estimate of the current level of the time series. This estimate is comprised of a weighted average of current and past values, where most weight is given to the most recent observations (with decreasing weight.....read more
Sampling (17)
17. Sampling
There are various ways of sampling a population, according to research and analysis goals.
17.1. Stratified Sampling
Stratified sampling involves breaking the population into strata (a.k.a. subgroups) according to a specific identifiable characteristic in such a way that each member of the population belongs to only one strata. Stratified random sampling is the process of selecting independent simple random samples from each strata. A question that arises here Is how to allocate the sampling effort among the strata. There are various possibilities:
- Proportional allocation: The proportion of the sample from a stratum is the same as the proportion of that stratum to the population. This is used if there is little to nothing known about the population and there are no strong requirements for the production of information.
- Optimal allocation: More sample effort is allocated to strata with a higher population variance. This is used if the objective is to estimate an overall population parameter (such as mean, total, or proportion) as precisely as possible. This method is only optimal with this goal in mind.
Analysing the results of stratified random samples is relatively straightforward, and any stratum sample mean (mj) can be used as an unbiased estimator of the population mean (μj). It can also be sued to estimate the population total, as this is the product of the population mean and the number of population members.
17.2. Other Ways to Sample
Various other sampling methods are:
- Cluster Sampling: This method can be used when a population can be subdivided into small geographical units, or clusters. A simple random sample of clusters is then selected, and each member of these clusters is contacted for data. Using this method very little prior information of the population is needed.
- Two-Phase Sampling: In this method the regular data-collection is preceded by a smaller pilot study, in which a smaller sample is used. This cost more time but allows for methods and procedures to be improved, and can provide some estimations for the true study.
- Non-random sampling: There are two main methods:
- Non-probabilistic sampling: Sample members are selected by convenience. This often means that the sample is not representative of the population and lacks proper statistical validity.
- Quota sampling: There are specified numbers of people of certain characteristics (race, age, gender etc.) that are contacted. This usually produces quite accurate estimates of population parameters, but it is not possible to determine the reliability of these estimates, because the sample was not randomly chosen.
Bullets Statistics for Business and Economics
12. Multiple Regression
- Regression objectives are either to predict the value of the dependent variable, or to estimate the marginal effect of each independent variable.
- A population multiple regression model is a model that includes multiple independent variables.
- Standard multiple regression assumptions include the four standard simple regression assumptions, plus a fifth one: It is not possible to find a set off nonzero numbers such that the sum of the coefficients equals zero.
- Multiple regression models include an error term, ε, that represents variability caused by variables not included in the model.
- In multiple regression coefficients are estimated using least squares, but these estimates become less reliable the higher the correlations between independent variables are.
- Any regression coefficient in a multiple regression model is dependent on all independent variables, and are thus referred to as conditional coefficients.
- Mean square regression (MSR) shows the proportion of the variability by the dependent variable that can be explained by the regression model.
- In a multiple regression model the sum-of-squares (SST; or sample variability) can be split into the sum of squares regression (SSR; or explained variability) and the sum of squares error (SSE; or unexplained variability). This is referred to as sum-of-squares decomposition.
- The coefficient of determination, R2, describes the strength of the linear relationship between the independent variables and the dependent variables, and is calculated by 1 – SSE/SST.
- Adding more independent variables leads to a misleading increase in R2, which can be avoided by calculating the adjusted coefficient of determination.
- The coefficient variance estimator, s2b, is calculated as:
The square root of s2b is the coefficient standard error.
- Multiple regression models can be transformed into non-linear models, namely quadratic models and logarithmic models.
- Dummy variables can be used to represent categorical data in a regression model, and have a value of either 0 or 1.
13. Additional Topics in Regression Analysis
- Models are developed through four steps: model specification (selecting the variables, the algebraic form, and the data), coefficient estimation, model verification (checking whether the model is still accurate), and interpretation and inference.
- Dummy variables can be used to represent more than two categories by using multiple dummy variables. The rule is: number of categories -1 = number of dummy variables.
- In time series data the values of the dependent variable are related, this is then referred to as a lagged dependent variable.
- Not including important independent variables in a model can make any conclusions drawn from this model faulty.
- Multicollinearity is the phenomenon of two highly correlated independent variables. This leads to misleading estimated coefficients.
- Correlations between error terms are called auto-correlated errors. This leads to the estimated standard errors for the coefficients being biased, the null hypotheses falsely being rejected, and confidence intervals being too narrow. Autocorrelation can be formally tested with the Durbin-Watson test.
15. Analysis of Variance
- An Analysis of Variance (ANOVA) can be used to analyze data at more .....read more
Oefenvragen Statistics for Business and Economics
12. Multiple Regression
1. A study was conducted to assess the influence of various factors on the start of new firms in the agricultural industry. For a sample of 70 countries the following model was estimated:
yn = -59.31 + 4.983x1 + 2.198x2 + 3.816x3 - 0.310x4 11.1562 10.2102 12.0632 10.3302
-0.886x5 + 3.215x6 + 0.85x7 13.0552 11.5682 10.3542
R2 = 0.766
where:
yn = new business starts in the industry
x1 = population in millions
x2 = industry size
x3 = measure of economic quality of life
x4 = measure of political quality of life
x5 = measure of environmental quality of life
x6 = measure of health and educational quality of life
x7 = measure of social quality of life
The numbers in parentheses under the coefficients are the estimated coefficient standard errors.
a. Interpret the estimated regression coefficients.
b. Interpret the coefficient of determination.
c. Find a 90% confidence interval for the increase in new business starts resulting from a one-unit increase in the economic quality of life, with all other variables unchanged.
d. Test, against a two-sided alternative at the 5% level, the null hypothesis that, all else remaining equal, the environmental quality of life does not influence new business starts.
e. Test, against a two-sided alternative at the 5% level, the null hypothesis that, all else remaining equal, the health and educational quality of life does not influence new business starts.
f. Test the null hypothesis that, taken together, these seven independent variables do not influence new business starts.
2. Based on 25 years of annual data, an attempt was made to explain savings in Japan. The model fitted was as follows:
y = b0 + b1x1 + b2x2 + e
where
y = change in real deposit rate
x1 = change in real per capita income
x2 = change in real interest rate
The least squares parameter estimates (with standard errors in parentheses) were (Ghatak and Deadman 1989) as follows:
b1 = 0.097410.02152 b2 = 0.37410.2092
The adjusted coefficient of determination was as follows:
R2 = .91
a. Find and interpret a 99% confidence interval for b1.
b. Test, against the alternative that it is positive, the null hypothesis that b2 is 0.
c. Find the coefficient of determination.
d. Test the null hypothesis that b1 = b2 = 0.
e. Find and interpret the coefficient of multiple correlation.
3. Based on data from 63 countries, the following model was estimated by least squares:
yn = 0.58 - .052x1 - .005x2 R2 = .17
1.0192 1.0422
where:
yn = growth rate in real gross domestic product
x1 = real income per capita.....read more
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