**CHAPTER ONE**

**INTRODUCTION**

_{1}and another n

_{2}n

_{1}consist of those consumers about whom we have information on the regressors (say income, mortgage interest rate, number of people in the family etc) as well as the regressed (amount of Expenditure on housing).

_{2 }consumers about whom we have information only on the repressors but not on the regressed.

_{1}if y is not observed (because of censoring) all such observations (=n

_{2}) denoted by crosses will lie on the horizontal axis

_{1}) denoted by dots will lie in the X – Y plane.

_{1}X

_{i}+ U

_{i}if RHS>0 = 0

_{1}observations only, the resulting intercept and slope coefficient are bound to be different than if all the (n

_{1}+ n

_{2}) observations were taken into account.

**CHAPTER TWO**

**ILLUSTRATION OF TOBIT MODEL**

_{i}. This variable linearly depends on T

_{i}via a parameter (vector) which determines the relationship between the independent variable (or vector) T

_{i}and the latent variable y

_{i}(just as in a linear model). In addition, there is normally distributed error term U

_{i}to capture random influences on this relationship. The observable variable y

_{i}is defined to be equal to the latent variable whenever the latent variable is above zero and zero otherwise.

_{i}= y

_{i}if y

_{i}> 0

_{i}≤ 0

_{i}is a latent variable:

_{i}= Bx

_{i}+ u

_{i}, u

_{i }N(0, o

^{2})

**Consistency**

^{B}is estimated by regressing the observed y

_{ion}x

_{i}, the resulting ordinary least squares regression estimator is inconsistent. It will yield a downwards-based estimated of the slop coefficient and an upwards-biased estimated of the intercept. Takeshi Amemiya (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.

**Interpretation**

^{B}coefficient should not be interpreted as the effect of x

_{ion}y

_{i}, as one would with a liner regression model; this is a common error. Instead, it should be interpreted as the combination of (1) the change in y

_{i}of those above the limit, weighted by the probability of being above the limit; and (2) the change in the probability of being above the limit, weighted by the expected value of y

_{i}if above.

^{(2)}

**Variations of the Tobit model**

**Type 1**

_{i}cannot always be observed while the independent variable x

_{i}is observable. A common variation of the Tobit model is censoring at a value y

_{L}different from zero:

_{i}= y

_{i}if y

_{i}> y

_{L}

_{L}if y

_{i}≤ y

_{L}

_{i}= y

_{i}if y

_{i}> y

_{u}

_{u}if y

_{i}≤ y

_{u}

_{i}is censored from above and below at the same time.

_{i}= y

_{i}if y

_{i}> y

_{L}< yu

_{L}if y

_{i}≤ y

_{L}

_{u}if y

_{i }≤ yu

_{i}= y

_{2i}if y

_{i}> 0

_{i}≤ 0.

**Example 1:**Consider the situation in which we have a measure of academic aptitude (scaled 200-800) which we want to model using reading and math test scores, as well as, the type of program the student is enrolled in (academic, general, or vocational). The students who answer all questions on the academic aptitude test correctly receive a score of 800, even though it is likely that these students are not “truly” equal in aptitude. The same is true of students who answer all of the questions incorrectly. All such students would have a score of 200 (i.e. the lowest score possible). Meaning that even though censoring from below was possible, but it does not occur in the dataset.

**HYTHESIS TESTING**

**What is Hypothesis Testing?**

**Types of hypothesis**

_{i}or Ha, it is the hypothesis (H

_{i}is that sample observations are influenced) a statement that by some non-random cause directly contradicts a null hypothesis for example, suppose we wanted to determine by stating that whether a coin was fair and balanced. The actual value a null hypothesis might be that half of the of a population flips would result in heads and half in parameters is Tails. The alternative hypothesis might be less than, (<) that the number of the leads and Tails greater than> or would be very different. Symbolically, not equal to (=) these hypothesis would be expressed as the value stated in the null H0: p = 0.5

_{i}) is the statement that directly contradicts a null hypothesis by stating that the actual value of a population parameter is more than (>) or less than (<) or not equal to (=) the value stated in the null hypothesis. The alternative hypothesis states what we think wrong about the null hypothesis.

**Four steps in hypothesis**

**Can Null Hypothesis Be Accepted**

**Decision Errors**

**Decision rules**

**P-value:**The strength of evidence in support of a null hypothesis is measured by the P-value. Suppose the test statistic is equal to S. the P=value is the probability of observing a test statistic as extreme as S, assuming the null hypothesis is true. If the P-value in less than the significance level, we reject the null hypothesis.

**One Tailed and Two Tailed Tests.**

**REFERENCES**

*“The Uses of tobit Analysis”, The Review of Economics and Statistics*(The MIT Press) 62 92): 318-321