Title

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What factors improve the chances of making a merger successful?Have you ever thought that

Can we increase the probability of making a company more financial sound by increase its Cash from Operating Activities to Sales?Have you ever thought that

What factors make a person to invest in Stock Market or not to invest in the Stock Market? Have you ever thought that

What kind of attributes contribute towards making an account a delinquent account? Have you ever thought that

Is it possible to identify underlying characteristics of companies which are responsible for putting them either at the top or at the bottom? Have you ever thought that

If you think that you are curious enough to look for answers these issues, then you have to equip yourself with ???

techniques that are capable of handlingDummy/Qualitative/Binary/Limited Dependent Variable Looking for models that have

Yi is a dummy variable.

The Liner Probability Model, the Probit Model and the Logit Model And, this takes us to

Topic for the day is

Lets consider a problemA researcher, Mr. Suyra Kant, wants to know what are the factors that determine the bankruptcy of a firm. He took a sample of 33 Solvent Firms and 33 Bankrupt Firms; and collected information on the following financial parameters:Working Capital to Total Assets (WC_ASSET)Retained Earnings to Total Assets (RE_ASSET)Return on Assets (ROA)Assets Turnover Ratio (ASSET_TURN)Market Value to Book Value (MV_BV)

Mr. Suyra Kant has made the following model

Binary Variable; GROUP=0, the company is bankrupt and GROUP=1, the company is solvent.What does this equation means? If the estimated value of the dependent variable GROUP is 0.75, how to interpret it?

To appreciate the issues involved in a regression model with binary dependent variable, lets consider a model with one explanatory variableWe take a general model as thus:

Assume that OLS is used to estimate the above model, then how to interpret the estimated value of Yi?

E(Yi|Xi) is nothing but expected value of Yi given Xi.

continued Alternatively, the expectation of Yi given Xi can also be calculated as thus:Let pi be the probability that a Yi takes a value 1 given a particular Xi and (1-pi) be the probability that a Yi takes a value 0 given a particular Xi. Then, the expected value of Yi given Xi can be calculated as thus:

continued It is nothing but the estimated value of the following model -

Thus, the estimated value is interpreted as the probability of having Yi is as one. The interpretation of the coefficient of Xi is marginal increase in the Probability that a particular Y takes a value one for one unit increase in X.

Therefore, such a model is called the LINEAR PROBABILITY MODEL.

continued We may have the graph of the estimated model as thus:

XXi10

y, pi

continued We may have the graph of the estimated model as thus:

XXi10

y, pi

AB

An ExampleAssume that the bankruptcy model is as thus:

As per the model, it assumed that bankruptcy depends upon the Return on Assets; and it is suggested that higher the Return on Assets a company generates, higher the probability that a company will be solvent. Using OLS, we obtain the results as thus:

An Example(continued)

Interpret the results?

An Example(continued)

If a firm has ROA = 20%, then its probability of being solvent is 55.25178%

Now, lets take the Complete ExampleThe Model as hypothesized by Mr. Surya Kant:

As per the model, it assumed that bankruptcy depends upon the financial ratios - Working Capital to Total Assets, Retained Earnings to Total Assets, Return on Assets, Assets Turnover and Market Value to Book Value. Using OLS, we obtain the results as thus:

The Complete Example(continued)Interpret the results?

The Complete Example(continued)Interpret the results?

Does this model create problems for us?@@@######

The Linear Probability Model Problem No. 1As per the Linear Probability Model, the probability increases with the values of X and it could be possible that at times, the values of probabilities are negative (1).In that case, it is difficult to interpret the results.

The Linear Probability Model Problem No. 1 (Example)

The Linear Probability Model Problem No. 2The Distribution of the Error term is not Normal Probability Distribution but it is Binomial Probability Distribution.

The Linear Probability Model Problem No. 3The error term are not homoscedastic but they have a problem of heteroscedasticity.

The Linear Probability Model Problem No. 4In the Linear Probability Model, as X increases the probability value continues to increase/decrease at a constant rate. Since the probability values should lie between 0 and 1, that is 0 < p < 1, a constant rate of increase/decrease is impossible.We need non-linear relation to restrict the probability values between 0 and 1.

How to ensure the choice of p within interval [0,1]?To limit the value of p within [0,1], usually S-Shaped relationship is used between x and p.In this regard, two widely used approaches exist one, that makes use of Logistic Function; and another, use Standard Normal Probability Distribution. By using the Logistic Function, we obtain a model which is called LOGIT MODEL; and the one which is obtained by using Standard Normal Probability Distribution is called PROBIT MODEL.

PROBIT AND LOGIT MODELSNORMAL PROBABILITY DISTRIBUTION LOGISTIC FUNCTION

PROBIT MODEL

LOGIT MODEL

Lets start with PROBIT Model Unlike the Linear Probability Model, the PROBIT Model assumes that the underlying distribution is Cumulative Standard Normal Probability Distribution whose shape is as follows:

Note that it is non-linear and its value lies between 0 and 1.

PROBIT ModelThe PROBIT Model expresses the probability p - a dependent variable Y takes the value 1 for given Xi.And, p is

Where the model is considered is having only one explanatory variable and the Function F represents Cumulative Normal Probability Distribution.Please note the density function of Standard Normal Probability Distribution -

PROBIT ModelIt is called Normal Equivalent Deviate (n.e.d) or Normit.

Estimation of PROBIT ModelThe PROBIT Model is not estimated by using OLS but it is estimated by using Maximum Likelihood Method and thus, the estimates we obtain are known as Maximum Likelihood Estimates.

Revisiting our Bankruptcy ExampleAssume that the bankruptcy model is as thus:

As per the model, it assumed that bankruptcy depends upon the Return on Assets; and it is suggested that higher the Return on Assets a company generates, higher the probability that a company will be solvent.

The Result obtained from Eviews

Z = -0.028116+0.110879 ROA

Now, lets take the Complete ExampleThe Model as hypothesized by Mr. Surya Kant:

As per the model, it assumed that bankruptcy depends upon the financial ratios - Working Capital to Total Assets, Retained Earnings to Total Assets, Return on Assets, Assets Turnover and Market Value to Book Value. Using the PROBIT Model, we obtain the results as thus:

The Result obtained from EviewsZ = -1.100487+0.117069 ROA+0.013031 MV_BV

Now, lets start with LOGIT Model The LOGIT Model assumes that the underlying distribution is a logistic function or Sigmoid Function whose shape is as follows:Note that it is non-linear and its value lies between 0 and 1.

LOGIT ModelThe LOGIT Model expresses the probability p that a dependent variable Y takes the value 1 given Xi.For the LOGIT Model, a particular type of logistic function is used which is called SIGMOID FUNCTION given below

Using the above function form, we may express the probability p that a dependent variable Y takes the value 1 given Xi assuming that there is only one explanatory variable.

LOGIT Model(continued)

This term is called LOGIT and the Model is known a LOGIT Model.

Finally, the LOGIT Model is

Estimation of LOGIT ModelThe LOGIT Model is not estimated by using OLS but it is estimated by using Maximum Likelihood Method and thus, the estimates we obtain are known as Maximum Likelihood Estimates.

Revisiting our Bankruptcy ExampleAssume that the bankruptcy model is as thus:

As per the model, it assumed that bankruptcy depends upon the Return on Assets; and it is suggested that higher the Return on Assets a company generates, higher the probability that a company will be solvent.

The Result obtained from EviewsZ = -0.180031+0.200139 ROA

Now, lets take the Complete ExampleThe Model as hypothesized by Mr. Surya Kant:

As per the model, it assumed that bankruptcy depends upon the financial ratios - Return on Assets, and Market Value to Book Value. Using the LOGIT Model, we obtain the results as thus:

The Result obtained from EviewsZ = -2.042474+0.218424 ROA+0.023347 MV_BV

Another ExampleA survey is conducted among the users and non-users of micro-wave oven. The following information is being collected about them on the following parameters:Users/Non-Users (Users-1 and non-users-0)Monthly Income (Rs. in 000)Monthly Consumption of Food Items (in Kgs.)

Data Collected is

Think???What should be the LOGIT MODEL?

The Result of the Estimated LOGIT Model

The Result of the Estimated LOGIT Model

Now, lets think about???What should be the PROBIT MODEL?

The Result of the Estimated PROBIT Model

The Result of the Estimated PROBIT Model

Using LOGIT and PROBIT modelscan we predict who is going to the OWNER of Microwave?

Yes, for that one has to take a threshold limit of the probability!

Normally, such a threshold is taken as p = 50%. If Probability is more than 50%, he is going to be the owner; otherwise he is not!

Can we use R2 or Adjusted R2 for determining Goodness of Fit?Answer is NO.Then, what should we use for Goodness of Fit?

Option #1: Use Log-likelihoodLog-likelihood Statistic can be used to gauge Goodness of Fit of Logit/Probit Models. It is defined as

Some software provide an estimate of this as (-2Log-likelihood) or -2LL to get a positive number. Lower -2LL, better is the fit.

Option #2: Use Log-likelihood Ratio A -2Log-likelihood Ratio (-2LLR) can also be used to gauge Goodness of Fit of Logit/Probit Models. It is defined as

Where LLR the value of Log-likelihood from the restricted model (with only intercept) and LLU the value of Log-likelihood from the unrestricted model (one with all explanatory variables).

Note that -2LLR follows Chi-Square Distribution.61

If you can not use R-Square thenUSE Pseudo R-Square.There are several measures which are intended to mimic the R-Squared, but none of them are an R-Squared. Hence, such measures are called Pseudo R-Square.

Option #3: Use Pseudo R-SquareMcFaddens R-Square is one of the Pseudo R-Square measures. It is defined as

Its values lying between 0.2 to 0.4 are considered highly satisfactory.

Note that -2LLR follows Chi-Square Distribution.63

Option #4: Use Pseudo R-SquareAnother measure of Pseudo R-Square is Cox and Snell. It is also based on log-likelihood but it takes the sample size into account.It is defined as

Note that -2LLR follows Chi-Square Distribution.64

Option #5: Use Pseudo R-SquareAnother measure of Pseudo R-Square is Nagelkerke measure. It adjusts the Cox and Snell measure for the maximum value so that 1 can be achieved.

It is defined as -

Note that -2LLR follows Chi-Square Distribution.65

Lets take one more EXAMPLEA company is selling R. O. System for home under the brand of Puredews. Their marketing head was interested in predicting what is chance that a person is likely to use their products. For this purpose, a sample of 231 users of their R. O. System and 519 non-users of their R. O. System was taken with their monthly income (in thousands), Age and Family Size believing that these determines the uses and non-users.Data is given in an Eview File.

We obtain the following PROBIT Results

We obtain the following LOGIT Results

Which one should we use?PROBIT MODEL OR LOGIT MODEL?

LOGIT vs PROBITThe main difference is that the logistic distribution has comparatively slightly fatter tails as compared to that of Probit Model.

In majority of applications, both give the similar results. However, researchers and practitioners prefer to use LOGIT because of its comparative mathematical simplicity.

Now, the question is what is the relation between the Age, Size and Income and the Probability that a person selected is user of the Companys R. O. System?

First challenge in the issue is - relationship between the probability values and independent variablesThough there is a non-linear relation between the probability and an independent variable, yet the sign of coefficient shows the direction of relation between the probability and the independent variable.

Second challenge in the issue is What is the rate of change of probability due to change in independent variable?The rate of change is given as thus...

Go to EXCEL For further and better understanding

How do you feel about LOGIT and PROBIT?

THANK YOUVERY MUCH

CREDIT RATINGConsumerAnnual Income(Rs. in lakhs)Age(in Years)No. of ChildrenHIGH RISK09.20273010.7024008.90322011.2029409.90313010.7029108.6028109.10315010.30262010.50303LOW RISK118.6423117.4475122.6411124.3390119.4432114.2463112.7424121.6482126.4373119.4511

USER OF MICRO-WAVEConsumerMonthly Income(Rs. '000)Monthly Consumption of Food Items(in Kgs)Dependent Variable: USER_MICRO_WAVEUSERS OF MICRO-WAVE115.0062Method: ML - Binary Logit123.5054Date: 06/16/10 Time: 16:10116.6078Sample: 1 24115.5074Included observations: 24124.0088Convergence achieved after 6 iterations131.7066Covariance matrix computed using second derivatives131.0058VariableCoefficientStd. Errorz-StatisticProb.122.6082C-18.492688.182243-2.2600990.0238118.0070INCOME0.3325760.1628922.0416890.0412126.0074CONSUMPTION_FOOD0.1927560.0925592.0825270.0373112.0080Mean dependent var0.5S.D. dependent var0.510754122.0070S.E. of regression0.350608Akaike info criterion0.88847NON-USERS OF MICRO-WAVE020.068Sum squared resid2.581446Schwarz criterion1.035727012.674Log likelihood-7.66164Hannan-Quinn criter.0.927537016.656Restr. log likelihood-16.63553Avg. log likelihood-0.31923509.472LR statistic (2 df)17.94778McFadden R-squared0.539441023.058Probability(LR stat)0.000127011.458Obs with Dep=112Total obs24014.850Obs with Dep=012017.062010.85206.064If a person has -012.040Monthly Income =12.00016.044Consumption of Food Items =75.00thenthe Value of Z will be-0.045068The Probability that he is USER is =48.87%Dependent Variable: USER_MICRO_WAVEMethod: ML - Binary ProbitDate: 06/16/10 Time: 16:36Sample: 1 24Included observations: 24Convergence achieved after 6 iterationsCovariance matrix computed using second derivativesVariableCoefficientStd. Errorz-StatisticProb.C-10.812934.550141-2.3763950.0175INCOME0.1932250.0900562.145620.0319CONSUMPTION_FOOD0.1133030.0529922.1381210.0325Mean dependent var0.5S.D. dependent var0.510754S.E. of regression0.349943Akaike info criterion0.879612Sum squared resid2.571666Schwarz criterion1.026868Log likelihood-7.55534Hannan-Quinn criter.0.918679Restr. log likelihood-16.63553Avg. log likelihood-0.314806LR statistic (2 df)18.16038McFadden R-squared0.545831Probability(LR stat)0.000114Obs with Dep=112Total obs24Obs with Dep=012If a person has -Monthly Income =12.00Consumption of Food Items =75.00thenthe Value of Z will be0.003495The Probability that he is USER is =50.14%

Sheet2Dependent Variable: USER_MICRO_WAVEMethod: ML - Binary LogitDate: 06/16/10 Time: 16:10Sample: 1 24Included observations: 24Convergence achieved after 6 iterationsCovariance matrix computed using second derivativesVariableCoefficientStd. Errorz-StatisticProb.C-18.492688.182243-2.2600990.0238INCOME0.3325760.1628922.0416890.0412CONSUMPTION_FOOD0.1927560.0925592.0825270.0373Mean dependent var0.5S.D. dependent var0.510754S.E. of regression0.350608Akaike info criterion0.88847Sum squared resid2.581446Schwarz criterion1.035727Log likelihood-7.66164Hannan-Quinn criter.0.927537Restr. log likelihood-16.63553Avg. log likelihood-0.319235LR statistic (2 df)17.94778McFadden R-squared0.539441Probability(LR stat)0.000127Obs with Dep=112Total obs24Obs with Dep=012

Sheet3