The document discusses non-linear regression techniques for fitting curved relationships to data. It describes how some non-linear equations, such as exponential growth and constant elasticity models, can be transformed into linear equations suitable for least squares regression by taking the natural logarithm of variables. This allows the curved relationships to be estimated while maintaining the assumptions of linear regression. Examples are provided to illustrate how to perform the transformations and interpret the results.
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
Introduces and explains the use of multiple linear regression, a multivariate correlational statistical technique. For more info, see the lecture page at http://goo.gl/CeBsv. See also the slides for the MLR II lecture http://www.slideshare.net/jtneill/multiple-linear-regression-ii
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
Introduces and explains the use of multiple linear regression, a multivariate correlational statistical technique. For more info, see the lecture page at http://goo.gl/CeBsv. See also the slides for the MLR II lecture http://www.slideshare.net/jtneill/multiple-linear-regression-ii
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Two-Variable (Bivariate) RegressionIn the last unit, we covered LacieKlineeb
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Two-Variable (Bivariate) Regression
Regression uses the equation of a line to estimate the relationship between x and y. You may remember back in algebra learning about the equation of a line. Some learned it as Y =s X + K or Y = mX + B. In statistics, we use a different form:
Equation 1: Y = B0 + B1X + u
Let’s define each term in the equation:
· Y is the dependent variable. It is placed on the Y (vertical) axis. In the example below, the dependent variable (Y) is the infant mortality rate.
· B0 is the Y intercept. B0 is also referred to as “the constant.” B0 is the point where the regression line crosses the Y axis. Importantly, B0 is equal to the
predicted value of Ywhen X=0. In most cases, B0 is does not get much attention for two reasons. First, the researcher is usually interested in the relationship between x and y. not the relationship between x and y at the single value of x=0. Second, often independent variables do not take on the value zero. Consider the AECF sample data. There are no states with low-birth-weight percentages equal to zero, so we would be extrapolating beyond what the data tell us.
· B1 is usually the main point of interest for researchers. It is the slope of the line relating x to y. Researchers usually refer to B1 as a slope coefficient, regression coefficient or simply a coefficient.
B1 measures the change in Y for a one-unit change in x. We represent change by the symbol ∆.
B1 =
· u is the error term. The error term is the distance between the regression line and the dots on the scatterplot. Think about it, regression estimates a single line through the cloud of data. Naturally, the line does not hit all the data points. The degree to which the line “misses” the data point is the error. u can also be thought of as
all the other factors that affect the infant mortality rate besides X. Importantly, we
assume that u is totally random given X.
The ...
For this assignment, use the aschooltest.sav dataset.The dMerrileeDelvalle969
For this assignment, use the aschooltest.sav dataset.
The dataset consists of Reading, Writing, Math, Science, and Social Studies test scores for 200 students. Demographic data include gender, race, SES, school type, and program type.
Instructions:
Work with the aschooltest.sav datafile and respond to the following questions in a few sentences. Please submit your SPSS output either in your assignment or separately.
1. Identify an Independent and Dependent Variable (of your choice) and develop a hypothesis about what you expect to find. (
note: the IV is a grouping variable, which means it needs to have more than 2 categories and the DV is continuous)
2. Run Assumption tests for Normality and initial Homogeneity of Variance. What are your results?
3. Run the one-way ANOVA with the Levene test & Tukey post hoc test.
a. What are the results of the Levene test? What does this mean?
b. What are the results of the one-way ANOVA (use notation)? What does it mean?
c. Are post hoc tests necessary? If so, what are the results of those analyses?
4. How do your analyses address your hypotheses?
Is concentration of single parent families associated with reading scores?
Using the AECF state data, the regression below measures the effect of the state's percentage of single parent families on the percentage of 4th graders with below basic reading scores.
%belowbasicread = β0 + β1x%SPF + u
Stata Output
1) Please write out the regression equation using the coefficients in the table
2) Please provide an interpretation of the coefficient for SPF
3) How does the model fit?
4) What is the NULL hypothesis for a T test about a regression coefficient?
5) What is the ALTERNATE hypothesis for a T test about a regression coefficient?
6) Look at the p value for the coefficient SPF.
a) Report the p value
b) How many stars would it get if we used our standard convention?
* p ≤ .1 ** p ≤ .05 *** p ≤ .01
image1.png
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Tw ...
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These slides review basic math tools used in our economics course: the concept of multi-variate relations and their mathematical representation as functions, bivariate functions, linear functions, etc.
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USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
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Empowering the Unbanked: The Vital Role of NBFCs in Promoting Financial Inclu...Vighnesh Shashtri
In India, financial inclusion remains a critical challenge, with a significant portion of the population still unbanked. Non-Banking Financial Companies (NBFCs) have emerged as key players in bridging this gap by providing financial services to those often overlooked by traditional banking institutions. This article delves into how NBFCs are fostering financial inclusion and empowering the unbanked.
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The Evolution of Non-Banking Financial Companies (NBFCs) in India: Challenges...beulahfernandes8
Role in Financial System
NBFCs are critical in bridging the financial inclusion gap.
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They support sectors like micro, small, and medium enterprises (MSMEs), housing finance, and personal loans.
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The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
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Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
2. NON-LINEAR REGRESSION 2
as ln(X), you can use the regular least squares method to fit the curve Y = AX to your data.b
Before we go further into how to use these new equations, we had better review logarithms and e.
Logarithms defined:
aIf a = c then b = log c .b
ab = log c is read, “b is the log to the base a of c.”
I remember the logarithm definition by remembering that:
The log of 1 is 0, regardless of the base.
Anything to the 0 power is 1.
If I can remember those two facts, then I can arbitrarily pick 10 as my base and write:
1010 = 1 and log 1 = 0.0
aFrom that I get a = c and log c = b, by substituting “a” for 10, “b” for 0, and “c” for 1.b
aYou can't take the log of 0 or of a negative number. The c in log c must be a positive number.
ee (. 2.718282) is the base of the “natural logarithms” (log is written “ln”).
Natural logarithms are used for continuous growth rates.
Something growing at a 100% annual rate, compounded continuously, will grow to e times its
original size in one year. (The diagram on the preceding page shows a 100% growth rate.)
Something growing at an r annual rate, compounded continuously, will grow to e times itsr
original size in one year.
In Excel, the function =EXP(A1) raises e the power of whatever number is in cell A1. =LN(A1) takes
ethe natural log of whatever is in cell A1.
Some exponential algebra rules
e = e ea+b a b
e = (e )ab a b
e = aln(a)
e = ab•ln(a) b
e = 10
Some corresponding logarithm algebra rules
ln( ab ) = ln( a )+ln( b )
ln( a ) = bAln( a )b
ln( e ) = aa
ln( 1 ) = 0
Non-linear models that can be made linear using logarithms:
Now we can get back to the curved models introduced at the beginning of this section and give fuller
explanations.
Constant growth or decay equation Y = Ae ubX
Suppose you’re modeling a process that you think has continuous growth at a constant growth rate. This is
like the diagram on page 1. You want to estimate the growth rate from your data.
3. NON-LINEAR REGRESSION 3
Y = Ae u is your equation.bX
X is often a measure of time. X goes up by 1 for each second, hour, day, year, or whatever time unit you
are using. Sometimes, X is a measure of dosage.
A is the starting amount, the amount expected when X is 0.
b is the relative growth rate, with continuous compounding.
u is the random error. Notice that we multiply by the error, rather than adding or subtracting the error as in
standard linear regression. u has a mean of 1 and is always bigger than 0. u is never 0 or negative.
Use this equation if:
The analogy with steady growth or decay makes sense.
Y cannot be 0 or negative.
X can be positive, 0, or negative.
The interpretation of b in Y = Ae u:bX
b is the parameter in which you are most interested, usually. Your estimate of b is your estimate of the
relative change in Y associated with a unit change in X.
Mathematically, if X goes up by 1, Y is multiplied by e . That is because Ae equals Ae , whichb b(X+1) bX+b
equals Ae e , which is Y is multiplied by e . That may not look like “relative change,” but it is, if you arebX b b
using continuous compounding.
People who use these models usually make a simpler statement about b. They say that if X goes up by 1,
Y goes up by 100b percent. They mean that if X goes up by 1, Y is multiplied by (1+b). For example, if
your estimate for b is 0.02, then it is said that if X goes up by 1, Y goes up by 2%, because Y is multiplied
by 1.02. (1 + .02 = 1.02.)
The 1+b idea is fine so long as b is near 0. For small values of b, e is close to 1+b. Multiplying by 1+b isb
close to the same as multiplying by e . For example, if b is 0.05 and X increases by 1, Y is multiplied byb
e , which is about 1.051, so Y really increases by 5.1%. That is pretty close to 5%. Most things in the.5
financial world and the public health world grow pretty slowly, so the 1+b or b-percent idea is OK. The
downloadable file about time series has more details on this.
How to make the growth/decay equation linear:
To convert Y = Ae u to a linear equation, take the natural log of both sides:bX
ln(Y) = ln(Ae u) Apply the rules above and you get:bX
ln(Y) = ln(A) + bX + ln(u)
To implement this, create a new variable y = ln(Y). (The Y in the original equation is “big Y.” The new
variable is “little y.”)
Also, define v as equaling ln(u) and a as equaling ln(A).
4. NON-LINEAR REGRESSION 4
Subsitute y for ln(Y), a for ln(A), and v for ln(u) in ln(Y) = ln(A) + bX + ln(u) and you get:
y = a + bX + v
This is a linear equation! It may seem like cheating to turn a curve equation into a straight line equation
this way, but that is how it’s done.
Doing the analysis this way is not exactly the same as fitting a curve to points using least squares. This
method this method minimizes the sum of the squares of ln(Y) - (ln(A) + bX), not the sum of the squares of
Y - Ae . In our equation, the error term is something we multiply by rather than add. Our equation isbX
Y=Ae u, rather than Y = Ae +u.bX bX
We make our equation Y=Ae u, rather than Y = Ae +u, so that when we take the log of both sides we getbX bX
an error, which we call v, the logarithm of u in the original equation, that behaves like errors are supposed
to behave for linear regression. For example, for linear regression we have to assume that the error’s
expected value is 0. To make this work, we assume that the expected value of u is 1. That makes the
expected value of ln(u) equal zero, because ln(1) = 0. In Y=Ae u, u is never 0 or negative, but v can takebX
on positive or negative values, because if u is less than 1, v=ln(u) is less than 0.
When you transform Y=Ae u to y = a + bX + v and do a least squares regression, here is how youbX
interpret the coefficients:
b8 is your estimate of the growth rate. a8 is your estimate of ln(A). To get an estimate of A, calcualte e to the
a-hat power.
Getting a prediction for Y from your prediction for y
After you calculate a prediction for little y, you have to convert it to big Y.
Your prediction for Y = e . In Excel, use the =EXP() function.Your prediction for y
An example
The first two columns are your data in a dose-response experiment. The third column is calculated from
the second by taking the natural logarithm.
Dose Bacteria LnBacteria
1 35500 10.4773
2 21100 9.95703
3 19700 9.88837
4 16600 9.71716
5 14200 9.561
6 10600 9.26861
7 10400 9.24956
8 6000 8.69951
9 5600 8.63052
10 3800 8.24276
5. NON-LINEAR REGRESSION 5
11 3600 8.18869
12 3200 8.07091
13 2100 7.64969
14 1900 7.54961
15 1500 7.31322
Your regression results are:
LnBacteria is the dependent variable, with a mean of 8.8309284
Variable Coefficient Std Error T-statistic P-Value
Intercept 10.57833 0.0597781 176.95997 0
Dose -0.2184253 0.0065747 -33.222015 0
You have found that each dose reduces the number of bacteria by about 22%.
Prediction, showing how it is calculated:
Variable Coefficient * Your Value = Product
Intercept 10.57833 1.0 10.57833
Dose -0.2184253 16.0 -3.4948041
Total is the prediction: 7.0835264
90% conf. interval is 6.861792 to 7.3052607
Your prediction for the number of bacteria if the dose is 16 is e = 1192. In Excel, calculate7.0835264
=EXP(7.0835264). The 90% confidence interval for that prediction is from e to e , which is6.861792 7.3052607
about 955 to 1488.
6. NON-LINEAR REGRESSION 6
Graph of Y=X u2
u is log-normally distributed with
a mean of 1.
b<1 example: Y = 5x u-1
Constant elasticity equation Y=AX ub
Another non-linear equation that is commonly used is the
constant elasticity model. Applications include supply,
demand, cost, and production functions.
Y = AX u is your equationb
X is some continuous variable that’s always bigger than 0.
A determines the scale.
b is the elasticity of Y with respect to X (more on elasticity
below).
u , the error, has a mean of 1 and is always bigger than 0.
Use this equation if:
• X and Y are always positive. X and Y can not be 0 or
negative.
• You suspect, or wish to test for, diminishing (|b|<1) or increasing (|b|>1) returns to scale
• If it is reasonable that the percentage change in Y caused by a 1% change in X is the same at any
level of X (or any level of Z, if you are doing a multiple regression). That is why this is called a
“constant elasticity” equation.
A constant elasticity model can never give a negative prediction. If a negative prediction would make
sense, you do not want this model. If a negative prediction would not make sense, this model may be what
you want.
The interpretation of b, the elasticity:
b is the percentage change in Y that you get when X changes by 1%. Economists call b the elasticity of Y
with respect to X.
If b is positive, you see a pattern like in the diagram above.
If b is negative in this equation, the pattern is like a hyperbola. The points
get closer and closer to the X axis as you go the right. The points get closer
and closer to the Y axis and they get higher and higher as X goes to the left
and gets near 0.
If you do a multiple regression version of this, with the equation Y =
AAX AZ Au, then b is the percentage change in Y that you get when X changesb c
by 1% and Z does not change. Economists say b and c are elasticities of Y
with respect to X and Z.
How to make this linear:
If you take the log of both sides of Y = AAX Au , you get:b
7. NON-LINEAR REGRESSION 7
ln(Y) = ln(A) + bAln(X) + ln(u)
To make this linear, create two new variables, y = ln(Y) and x = ln(X). (If you have more than one X
variable, take the logs of all of them.) Let a be ln(A) and v be ln(u), and our equation becomes nice and
linear:
y = a + bx + v
As with the growth model, to use least squares, we must assume that v behaves like errors are supposed to
behave for linear regression. One of those assumptions is that v’s expected value is 0. That is why we
assume that the mean of u is 1. That fits because ln(1) = 0. u is never 0 or negative, but v can take on
positive or negative values, because if u is less than 1, v=ln(u) is less than 0.
The multiple regression version of the constant elasticity model works like this:
Y = AAX AZ Au is the equation.b c
Take the log of both sides to get:
ln(Y) = ln(A) + bAln(X) + cAln(Z) + ln(u)
which you estimate as
y = a + bAx + cAz + v
where y = Ln(Y), a = ln(A), x = ln(X), etc.
This equation lets you look for diminishing or increases returns to scale. If |b+c| < 1, you have diminishing
returns. Doubling X and Z makes Y go up less than twofold. If |b+c|>1, you have increasing returns.
Doubling X and Z makes Y more than double.
Sometimes a linear model shows heteroskedasticity -- apparently unequal variances of the errors -- that
takes the form of the residuals fanning out or in as you look down the residual plot. This constant elasticity
form can correct that.
Getting a prediction for Y from your prediction for y
As with the growth-decay model, when you calculate a prediction for y, you have to convert it to Y. You
do this by raising e to the y power, because Y = e . In Excel, use the =EXP() function.y
An example
Here are some data:
Price Visits LnPrice LnVisits
10 512 2.30259 6.23832
8. NON-LINEAR REGRESSION 8
10 396 2.30259 5.98141
10 456 2.30259 6.12249
10 525 2.30259 6.2634
20 347 2.99573 5.84932
20 377 2.99573 5.93225
20 423 2.99573 6.04737
20 330 2.99573 5.79909
30 366 3.4012 5.90263
30 335 3.4012 5.81413
30 301 3.4012 5.70711
30 309 3.4012 5.73334
40 345 3.68888 5.84354
40 320 3.68888 5.76832
40 325 3.68888 5.78383
40 285 3.68888 5.65249
The first two columns are the actual data. The third and fourth columns are calculated as the natural logs
of the first two columns. 3.6888 approximately = LN(40), for example.
LnVisits is the dependent variable, with a mean of 5.9024412
Variable Coefficient Std Error T-statistic P-Value
Intercept 6.8044237 0.1496568 45.466851 0
LnPrice -0.2912347 0.047653 -6.1115685 0.0000269
The estimated coefficient of LnPrice says that when the price is 1% higher, there are 0.29% fewer visits.
To predict visits if the price is $25, first calculate =EXP(25). It is about 3.218875825. Use that in the
prediction form.
Prediction, showing how it is calculated:
Variable Coefficient * Your Value = Product
Intercept 6.8044237 1.0 6.8044237
LnPrice -0.2912347 3.2188758 -0.9374482
Total is the prediction: 5.8669755
90% conf. interval is 5.6865181 to 6.0474328
The prediction for the logarithm of visits is 5.8669755. The prediction for visits is =EXP(5.8669755),
which is about 353. The 90% confidence interval for that prediction is from =EXP(5.6865181) to
=EXP(6.0474328), which is about 295 to 423. Notice that 353 is not in the middle of this confidence
interval.
9. NON-LINEAR REGRESSION 9
More comments on our two non-linear models
Time series often use the ln(Y) = ln(A) + bX + ln(u) form.
The multiple regression version has more than one X variable. Your independent (X) variables
measure time, either
Sequentially: 0,1,2,3,... or 1996, 1997, 1998 , ... or
Cyclically: seasonal dummy variables (to be explained next week)
Notice that you do not take the logs of these X variables, just the Y variable.
The coefficient b is the growth/decline rate, meaning that a change in X of 1 corresponds with a
100b percent change in Y, if b is small.
Cross section studies, that look for causation rather than time trends, often use the constant elasticity form,
ln(Y) = ln(A) + bAln(X) + ln(u)
For this model, your independent variables have a proportional effect on your Y variable.
You do take the logs of your X variables.
Coefficients are elasticities, meaning that a change in X by 1% (of X) changes Y by b percent.
General comments on working with transformed equations
You must assume that the error in the transformed equation has the desired properties (normal distribution
with mean 0).
Once you get your estimates from the transformed equation, going back to the original equation can be
tricky. Some original-equation parameter estimates are biased, but consistent, if the parameter was
transformed (e.g. A in the models above). Confidence intervals around predicted values are no longer
symmetrical. You must get the confidence interval from the transformed equation and then transform the
ends back.
Before you start with using a non-linear regression equation:
Have a good reason for not using a linear model, such as
a theory of how the process that you're observing works, or
a pattern you see on a graph or in the residuals from a linear regression
Determine which non-linear equation is best for your data. Can you make a sensible analogy with
steady growth or decay? How about with demand or production?
To do the regression:
Use algebra (or just read the above) to transform your original non-linear equation into one that is
linear. For example, Y = AX u transforms tob
ln(Y) = ln(A) + bAln(X) + ln(u).
Create the variables you need from your data.
10. NON-LINEAR REGRESSION 10
Use least squares, including the transformed variables in your data set.
Your predictions, and at least some of your estimated coefficients, will have to be transformed to
anti-logs to get back to the parameters in the original non-linear equation. In both of these
equations, the predicted value of ln(Y) must be transformed with the exp function to get a predicted
value for Y.