Describe the properties of Linear Functions.

Example 1 - Linear Housing Expenditure Function

Example 1 - part 2

Example 1 - part 2

Example 2 - Demand for CD’s

Let *grthemp* denote the proportionate growth in employment, at the county level, from 1990 to 1995, and let *salestax* denote the county sales tax rate, stated as a proportion. Intrepret the intercept and slope in the equation:

*grthemp* = .043 - .78 *salestax*

According to the equation, the constant .043 is the proportionate growth in employment where there are no taxes on sales. In other words, if there are not taxes on sales, the employment would grow by 4.3 percentage points.

The slope can be interpreted as follows: an increase (decrease) of the sales tax rates by one percentage point decreases (increases) the growth in employment by -0.78 percentages points.

Suppose that in a particular state a standardized test is given to all graduating seniors. Let *score* denote a student’s score on the test. Someone discovers that performance on the test is related to the size of the student’s graduating high school class. The relationship is quadratic:

*score* = 45.6 + .082 *class* - .000147 *class*^{2}

where *class* is the number of students in the graduating class.

1) How do you literally interpret the value 45.6 in the equation? By itself, is it of much interest? Explain.

The value 45.6 is the intercept in the equation, so it literally means that if class = 0 then the score is 45.6. Of course, class = 0 can never happen. So, by itself, 45.6 is not of much interest. But it must be accounted for to use the equation to obtain score for sensible values of class.

Suppose that in a particular state a standardized test is given to all graduating seniors. Let *score* denote a student’s score on the test. Someone discovers that performance on the test is related to the size of the student’s graduating high school class. The relationship is quadratic:

*score* = 45.6 + .082 *class* - .000147 *class*^{2}

where *class* is the number of students in the graduating class.

2) From the equation, what is the optimal size of the graduating class (the size that maximizes the test score)? (Round your answer to the nearest integer.) What is the highest achievable test score?

Compute the ﬁrst derivative of the function score with respect to the variable class, and set it equal to zero. Solving the equation for the variable class we get

class = .082/ (2 X 0.000147) = 278.91. The optimal number of students is 279. The largest achievable score is 45.6 + (.082 X 279) - [.000147 X (279)^{2]} = 57.04

Suppose that in a particular state a standardized test is given to all graduating seniors. Let *score* denote a student’s score on the test. Someone discovers that performance on the test is related to the size of the student’s graduating high school class. The relationship is quadratic:

*score* = 45.6 + .082 *class* - .000147 *class*^{2}

where *class* is the number of students in the graduating class.

3) Does it seem likely that *score* and *class* would have a deterministic relationship? That is, is it realistic to think that once you know the size of a student’s graduating class you know, with certainty, his or her test score? Explain.

Of course it is not at all realistic to think that a student’s score is determined only by the size of his or her graduating class. This would imply that all students graduating in the same year from the same high school would have the same score. The class size could affect the learning experience, but most likely in a very marginal way or not at all.

It is hard to believe that a deterministic relationship with any set of variables exists, because the score might be affected by plenty of factors that cannotbe accounted for, as intelligence, personal situation, interest for the subject, physical and emotional conditions during the test etc

Suppose that person A earns £35,000 per year and person B earns £42,000.

1) Find the exact percentage by which person B’s salary exceeds person A’s.

42,000−35,000 / 35,000

= 0.2

so B earns 20% more than A

Suppose that person A earns £35,000 per year and person B earns £42,000.

2) Now, use the difference in natural logs to find the approximate percentage difference.

Let log(·) denotes the natural logarithm, then

log(42,000)−log(35,000) = 0.1823

The approximate percentage difference is 18%

What is the meaning of ceteris paribus?

When using ceteris paribus in economics, one assumes that all other variables except those under immediate consideration are held constant. For example, it can be predicted that if the price of beef increases—ceteris paribus—the quantity of beef demanded by buyers will decrease.

What is the meaning of a causal effect?

Therefore, causal effect means that something has happened, or is happening, based on something that has occurred or is occurring. A simple way to remember the meaning of causal effect is: B happened because of A, and the outcome of B is strong or weak depending how much of or how well A worked.

Define cross sectional data.

Cross-sectional data are observations that come from different individuals or groups at a single point in time. If one considered the closing prices of a group of 20 different tech stocks on December 15, 1986, this would be an example of cross-sectional data

Define Data frequency.

The frequency of a particular data value is the number of times the data value occurs. For example, if four students have a score of 80 in mathematics, and then the score of 80 is said to have a frequencyof 4. The frequency of a data value is often represented by f.

Define non experimental data

Non-experimental research is the label given to a study when a researcher cannot control, manipulate or alter the predictor variable or subjects, but instead, relies on interpretation, observation ( aka observational data) or interactions to come to a conclusion

Define panel data

panel data or longitudinal data are multi-dimensional datainvolving measurements over time. Panel datacontain observations of multiple phenomena obtained over multiple time periods for the same firms or individuals

define pooled cross section

A data configuration where independant cross sections, usually collected at different points in time, are combined to produce a single data set.

Define time series data

A time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data.

What are logirithms and natural logirithms?

Logarithms are like the opposite of exponents. Just like how adding and subtracting are “opposite” operations (and multiply/divide). For example:

103 = 1000. 10 is the base, 3 is the exponent, 1000 is the result.

The opposite direction of this is:

log10(1000) = 3. Again 10 is the base (it should be subscripted), 1000 is the result, and 3 is the exponent. Except now you’re working backwards from the result.

Generally log() means base 10. If it’s not base 10, they’ll write what base it is in subscript. Natural log, or ln(), is a special log where the base is e.

Why do we use logirithms?

There are two main reasons to use logarithmic scales in charts and graphs.

The first is to respond to skewness towards large values; i.e., cases in which one or a few points are much larger than the bulk of the data.

The second is to show percent change or multiplicative factors

Coefficient of determination.

The coefficient of determination, R2, is used to analyze how differences in one variable can be explained by a difference in a second variable. For example, the timing of when a person gets pregnant has a direct relation to when they give birth.

More specifically, R-squared gives you the percentage variation in y explained by x-variables. The range is 0 to 1 (i.e. 0% to 100% of the variation in y can be explained by the x-variables.

Control variable

Controlled variables are quantities that a scientist wants to remain constant, and she or he must observe them as carefully as the dependent variable

Covariate

Is the same as an explanatory variable/independant variable.

It is a variable that is used to explain variation in the dependant variable.

Degrees of freedom

The number of observations minus the number of estimated parameters.

df = 5; a t stat would need to be at least 2.57 for a significant result. Now raise the df to 15; the t stat would have to be 2.13 for significance. The higher the degrees of freedom, the lower the threshold for a significant result.

Dependant variable

The variable to be explained in a multiple regression model ( and a variety of other models)

Error term

The variable in a simple or multiple regression equation that contains unobserved factors which affect the dependant variable. the error term might also include measurement erros in the observed dependant or independent variables

Gauss markov assumption

states that in a linear regression model in which the errors have expectation zero, are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares, provided it exists

Heteroskedasticity

The variance of the error term, given the explanatory variable(s), is not constant

Explained Sum of Squares (SSE)

The total sample variation of the fitted values in a mupltiple regression model

Ordinary least squares

A method for estmiating the parameters of a multiple linear regression model.

the OLS estimates are obtained by minimizing the sum of squared residuals

Sum of squared residuals (ssr)

In a multiple regression analysis, the sum of the squared OLS residuals across all observations

What is a % confidence interval

A % confidence interval is how sure you are that the true mean(average) of a population(group) lies between the two interval numbers

So a 95% confidence interval of (8, 12) means that you are 95% sure that the average of the population is between 8 and 12. You are 95% sure because 95% of all data points will be between 8 and 12.

95% confidence interval example

You’re doing a survey on something…let’s say favourite fruit. If you want to know what everyone’s answer from your kindergarten class is you could easily just ask each one of them. Then you’d have 100% confidence that you’ve got the right answer.

However, the bigger the population you want to ask…let’s say your whole town of 10,000 people…the harder it is to ask all those people!

So you take a sample. You try to pick a group that represents the whole town. The bigger the sample you take, the closer to having a good representation of everyone in town. But after a while, you start getting diminishing returns. Adding one more person to the sample doesn’t add as much as the person before, and so on.

But because you’re only talking to a few people out of the whole town, you can’t be 100% sure. Depending on how many people you talk to, you can say you’re 99% sure, or 95%, or any other number for that matter.

how to calculate the intercept?

The intercept (often labeled the constant) is the expected mean value of Y when all X=0.

T-value

the t-statistic measures how many standard errors the coefficient is away from zero. Generally, any t-value greater than +2 or less than – 2 is acceptable. The higher the t-value, the greater the confidence we have in the coefficient as a predictor. Low t-values are indications of low reliability of the predictive power of that coefficient.