econometrics of microfinance- multiple linear regression

Brooke Gallagher

Econ 441

10/21/10

The Econometrics of Third World Development:Microcredit Financing and its affect on GDP- Multiple Linear

Regression

I. Introduction

Microcredit financing institutions bring a world of banking and financial services to the

poor in developing countries that would not otherwise be available. MFI’s make credit, savings,

insurance, and fund transfers available for low income citizens with the lofty ideal that this will

help to lower the inequality in income distribution among a country. The following paper

analyzes this question through multiple linear regression and economic theory. For the purpose

of this paper, it is less important how microcredit financing works as opposed to if it works.

With the following economic concepts in mind, it is natural to hypothesize that

microcredit financing does make a difference on the level of income inequality or the GINI

coefficient. The only faulty assumptions of MFI’s I can deliberate on is the query on whether it’s

enough to make a substantial difference in a countries’ GDP, whether or not these institutions

reach out to enough of the population. Individually, MFI’s have been known to work but is there

substantial evidence that they make an impact on an entire society? I would assume that they do

by analyzing economic theory mainly based on the fact that the more businesses there are in

operation, the more people there are employed, and the more competitive the wages. But this

may be an idealistic slippery slope and what may actually be happening is a perpetuation of

poverty. The perpetuation occurs through MFI’s accruing funds for loans to poor individuals that

would possibly be better allocated toward expanding enterprises such that more jobs would be

1 | P a g e

created. This is plausible due to the fact that most microenterprises which were started up

through microloans only create jobs for about three to four people on average and one would

assume that a medium size enterprise would stunt that number by the tens or even hundreds

(Book Review, pg. 1).

II. Economic Theory

Economic theory provides us with evidence that implementing the availability of

financial services (financial deepening) will promote a stale economy to flourish. This is due to

the fact that when people have access to savings accounts composed of interest rates, they are

likely to save more and with less credit constraints people are led to higher productivity rates

through more investments. With access to insurance, your assets are protected and fund transfers

lead to safe fund mobility. All of this leads to business diversity which is imperative for

economic growth. The most important fact is that less credit constraints on low income people

leads to higher productivity, which should lead to less inequality by looking at the profit

maximizing model for wage. Comparing marginal product of labor to wage, you will find that as

people become more productive (due to access to financial services), their Marginal Product

increases, and thus, the wages the wealthy pay to the poor also increase, narrowing the income

gap.

2 | G a l l a g h e r

The graph at right¹ demonstrates the labor market in a perfectly competitive market where Marginal Revenue is actually equal to the Marginal Product of Labor. In perfect competition, MPL is set to equal Marginal cost to create the highest possible profit. As you can see, if MPL shifts to the right (do to more productive workers), then the wage will increase.

¹This graph is meant to be seen as a template and Pm has no meaning for our current concern. (found on google.com; web images)

With analysis of the exogenous growth model, or Solow Model, we can see what affects

long run economic growth based on individual factors. In the short run, the model tells us that

subsidies can affect the steady state level of output and that the rate of growth as the economy

converges to the steady state is determined by the rate of capital accumulation (which is

determined by the savings rate and capital depreciation). Consequently, with access to a savings

account where one was not before, one would predict that savings rates would increase and thus,

capital accumulation. That which follows, in the long run, the model tells us that a country with a

higher savings rate will experience higher growth. The model assumes that there can only be

long term growth based on technological progress and labor force growth or else the growth line

converges to the steady state level where the growth rate is constant times the population growth.

It follows that microcredit financing leads to a higher employment rate due to the mass amount

of micro loans given to entrepreneurs to begin microenterprises. Hence, a larger labor force

should lead to growth in the long run. A higher loan rate leads to more innovation which in turn

will lead to technological advances. The following describes a situation where savings rate is

increased and economic growth is not experienced:


Hypothetically, the savings rate, sy is increased

to s1 y when there is an increased abundance of

available financial services. This makes savings per worker greater than population growth plus

depreciation, (n+d )k , thereby increasing capital

accumulation shifting the steady state from point A to point B. This also increases output per

worker, y0to y1expanding the economy.

However, the model tells us that although the economy will initially expand, it will return back to its steady state of growth where output per worker is constant growing at a rate of n, the population growth. The economic growth rate will be the same, according to the model but there will be a permanent higher amount of capital and productivity per worker.

Found via google.com web images.

This preceding argument in the model might suggest that microfinance is not the best

solution to development issues when in actuality, it might merely be implicit in nature that GDP

is just not the best measure of its success (or failure) due to the conclusions drawn from the

model. However, the model does not take into account entrepreneurship which is a very large

economic force and what microloans are based on; therefore, I think it would still be not only

interesting but important to run this regression.

Many MFI’s focus in on women and some will only strictly lend to women, mainly

because women have a much greater pay back rate. This allows women to enter the workforce,

educating them, and empowering them. The empowerment of women leads to increased decision

making at the community and household level which leads to more equality in things like

education. A more educated population leads to a more advanced and economically stable

society through higher wages. Men tend to invest more in physical capital whereas women tend

to invest more in human capital like education for their children. One study found that household

consumption increased by .40 percent per capita when women received the loan and men’s

consumption increased by only .23 percent (Using Microcredit, pg 3). When consumption is

increased, so is GDP because consumption is a function of GDP (GDP= C + I + G + net Exports,

where C is consumption, I is Investment, and G is Government Spending).

In poor communities, having assets may be a much better savings method than actually

saving in a bank. This is where microcredit loans come in. The reasons for this are varied.

Beginning with the simple fact that most banks in cities are a far bus ride away from the more

rural areas, tax rates are especially high (some as much as 75%), poor people tend to lend from

family and friends and therefore if you have saved money, it is likely you will be asked to give it

away. On average, one is able to save about $.25 to the $1 in poor communities (Why


Microcredit Works, pg1). Thus, it is much safer for a member of the above mentioned

community to invest their money which is why microfinance can be extremely valuable to a

person drowning in poverty.

Furthermore, microcredit financing is not a charity. It is not a onetime hand out with an

exorbitant group interest rate meant to make a profit off of either. MFI’s continue to lead

instruction to borrowers and customers on successful business operations to ensure sustainability.

This is indeed a form of education and when a subset of a population is educated, so is the

greater population (due to being affected by those around them) and when a population is better

educated, they can advance as a society. I would say that education is the root of a strong

economy. When people are better educated, they can not only earn better wages, but they can

start making better choices regarding finance and the political economy leading to prosperity and

growth.

Other aspects of an economy that impact GDP are wide and varied. To narrow the list

down, I have come up with the most important variables which may affect a country’s GDP level

based off of theory. The first is a question of what type of political state the country is in. It’s

pretty much given that a state of political unrest can be entirely detrimental to a country and its

consumption or savings rate. If a citizen doesn’t trust their government, they are less likely to

save their money in a financial institution (including an MFI) and it has been seen that

consumption decreases. However, if the country is in a state of war, the economy is likely to be

booming because the government creates jobs and pours money into military and defense. More

government spending equates to a higher level of GDP by simply looking at the following

equation: GDP = C + I + G + net exports. Therefore, it highly depends on what form of unrest

the country is going through to see how it will affect GDP.


Private property rights are another very important aspect to an economy. With these

rights people are more likely to invest in things such as land. Property rights also create a sense

of independence and dignity. People are more likely to work hard to earn these types of rights

which will lead to things such as innovation. Private property rights include the right to

determine its use and rights to exchange or sell the item with a mutual agreement. “The

fundamental purpose of property rights, and their fundamental accomplishment, is that they

eliminate destructive competition for control of economic resources. Well-defined and well-

protected property rights replace competition by violence with competition by peaceful means

(“Property Rights,” Albert Alchian).”

The government’s role in education can also greatly impact a country. A country whose

government funds and enforces that all children attend school until the age of eighteen versus a

country who only supports education up to age five will possess very different outcomes:

economically, socially, and politically. One would assume through theory that the economic rate

of return on education is very high for multiple reasons. The first reason the rate of return on

education for an entire society is high is that it makes the workforce more competitive and thus

increases wages. The second reason is that it leads to more technological innovation which will

lead to greater efficiency and subsequently, greater economic output. Third, a higher educated

population will be healthier through knowledge of good nutrition and greater access to health

care which means they will live longer lives which consequently means that the society will be

more productive with higher incomes over a greater amount of time. Lastly, it has been shown

that education and crime are negatively correlated which means that less people imprisoned

results in less money spent on police force and imprisonment, thus a gain to the social welfare.


Another variable which may significantly impact GDP is the level of corruption in the

country. A country with high rates of corruption by government officials is less likely to have

trusting citizens. Less trust in one’s government has been shown to have an adverse effect on

savings, investment rates (which, as we can see by the equation of GDP, as stated previously),

and tax collection. Corruption means a country is experiencing things like: blackmail from the

police force, extortion, and tax evasion (which may be a circular affect and actually be caused by

a government’s corrupt ways).

III. Data Collection

I chose to analyze an equal number of six countries from six different regions including:

Africa, E. Asia and the Pacific, E. Europe and Central Asia, Latin America and the Caribbean,

the Middle East and N. Africa, and South Asia. I retrieved my data through the internet. The

quantity of microfinance loans was found on mixmarket.org and the GDP levels from the

worldbank.org. As for finding how many years MFI’s have been implemented in a particular

country, the sources are varied but primarily were found through a google.com search.

My first independent variable of concern to help explain y, GDP, isx1. It will be the

measure of prevalence of microfinance institutions in a country over a given amount of time by

quantifying each countries gross loan portfolio. This is the best measure of microfinance because

it tells us how active the microfinance branches are in banking. If I were to just look at the

number of microfinance branches per country, my results would be off due to the fact that some

branches are smaller than others. The data that I found was all of the reported gross product

loans.


A second independent variable of interest x2 , is a categorical dummy variable quantifying

whether or not the country was in political turmoil or war torn for the duration of time that

microcredit financing was available. I gave each state of being a number: 0 meaning the country

was in a state of peace, 1 meaning there was some political unrest and insecurity, 2 meaning the

country was in a state of cross-country war, and 3 meaning the country was involved in a civil

war. I think this variable is extremely important to look at because whether a country is in war or

the type of unrest they are experiencing can greatly impact a society based off of economic

theory. This data was found on mapreport.com.

Third, the variable x3is another dummy variable to describe if the country has individual

property laws or not. Property laws are the rights to hold land as your own personal asset and not

to be taken away by anybody including the government. Since I couldn’t find data on property

rights exclusively, I decided to look at the scores on gfmag.com regarding economic freedom

which encompasses property rights. A 0 would mean the government holds all rights to the

property or land that would otherwise be rightfully one’s own and a 100 means there is complete

economic freedom. Included in economic freedom is: property rights, regulatory efficiency, ten

levels of economic openness, and competitiveness. Afghanistan is omitted because there was not

enough reliable information to build data from. I would imagine that a 0 would have an adverse

affect on GDP and reduce it significantly.

A forth independent variable of interest, x4, is education. With this I decided to count the

average level of education in each country. I feel that this should have a substantial affect on

GDP due to economic theory as stated above. I feel that the intercept in the regression will be

fairly large to signal a large return on educational investment. However, because educational

policies change frequently in some countries (especially those in political unrest), we may not


get an accurate picture just by looking at their average education and literacy rate today because

education has a somewhat lag effect. Meaning, although a strict education system may have been

implemented a year ago, a country will not reap the rewards for, say, fifteen years or so. With

this, I feel like it will be a benefit to look at the variables: “education” and “political state” as an

interaction term to see how they interact together since we know there is some correlation

between the two. I found data for the average years of schooling for adults over fifteen on

nationmaster.com under “Duration of compulsory education,” meaning the number of years a

child must legally attend school.

Corruption is the fifth independent variable,x5, being another categorical dummy variable

based on the level of corruption. Zero being no corruption, or very little, all the way up to nine,

being the most corrupt. I feel it is very important to examine this variable because economic

theory suggests that corruption has an adverse affect on GDP. This is also an interaction term

withx1, the level of political turmoil. This data was found on:

http://www.nationmaster.com/graph/gov_cor-government-corruption.


http://www.nationmaster.com/graph/gov_cor-government-corruption

Graphs and summary stats:

0.00 5.00 10.00 15.000

1000

2000

3000

4000

5000

Gross Loan Portfolio (mil-lions) vs GDP Growth

Gross Loan Portfolio (millions)Linear (Gross Loan Portfolio (millions))

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.000

0.5

1

1.5

2

2.5

3

3.5Politics vs GDP Growth

PoliticsLinear (Politics)

0.002.00

4.006.00

8.0010.00

12.0014.00

0

20

40

60

80

Property Rights vs GDP Growth

PropLinear (Prop)


0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.0002468

1012 Educ. vs GDP Growth

EducLinear (Educ)

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.000123456789

Corruption vs GDP Growth

CorruptLinear (Corrupt)

Corrupt Educ Prop

Mean 7.371429 Mean 7.771429 Mean 55.17143Standard Error 0.1483 Standard Error 0.393403 Standard Error 1.060423Median 8 Median 8 Median 55.4Mode 8 Mode 8 Mode 61Standard Deviation 0.877353

Standard Deviation 2.327403

Standard Deviation 6.273547

Sample Variance 0.769748

Sample Variance 5.416807 Sample Variance 39.35739

Kurtosis 5.18587 Kurtosis 2.16011 Kurtosis 0.162973Skewness -1.93796 Skewness -0.95335 Skewness -0.01689Range 4 Range 11 Range 29.2Minimum 4 Minimum 0 Minimum 40Maximum 8 Maximum 11 Maximum 69.2Sum 258 Sum 272 Sum 1931Count 35 Count 35 Count 35


PoliticsGross Loan Portfolio

(millions) Change in GDP

Mean1.45714

3 Mean876.331

4 Mean2.71500

3

Standard Error0.19378

1 Standard Error222.207

1 Standard Error0.41871

4

Median 1 Median 269.8 Median1.68520

9

Mode 1 Mode #N/A Mode #N/AStandard Deviation

1.146423

Standard Deviation

1314.595

Standard Deviation

2.477148

Sample Variance

1.314286

Sample Variance

1728161

Sample Variance

6.136261

Kurtosis-

1.40246 Kurtosis2.65098

6 Kurtosis 8.37508

Skewness 0.11124 Skewness1.94510

5 Skewness2.60089

9

Range 3 Range 4596.4 Range12.7563

8

Minimum 0 Minimum 3.6 Minimum0.22240

4

Maximum 3 Maximum 4600 Maximum12.9787

9

Sum 51 Sum 30671.6 Sum95.0251

2

Count 35 Count 35 Count 35

IV. Econometrics

Correlation Matrix:

GDP 2

Gross Loan

Portfolio (millions) Politics Prop Educ Corrupt

GDP 2 1Gross Loan Portfolio (millions) 0.288406 1Politics -0.2242 -0.05825 1Prop 0.016902 -0.11566 -0.24718 1Educ 0.07104 -0.39699 -0.47777 0.436856 1


Corrupt -0.25046 0.02215 0.030913 -0.48642 -0.20206 1This matrix tells us that the following are positively correlated with a change in GDP: gross loan

portfolio, property rights, and education. Politics is negatively correlated to a change in GDP

meaning the more corrupt a government was, the less positive change in GDP they saw. The

following are negatively correlated with Microfinance Gross Loan Portfolio: politics and

property rights. Politics and property rights have a weak negative correlation and education and

politics has a fairly strong negative correlation. Education and property rights have a positive

correlation, but corruption and property rights have a negative correlation. All these correlations

coincide with economic theory.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.46841R Square 0.219408Adjusted R Square 0.084823Standard Error 311.7952Observations 35

ANOVA

df SS MS FSignificanc

e F

Regression 5792438.

1158487.

61.63025

9 0.183465

Residual 29281927

197216.2

3

Total 34361170

9

Coefficien

tsStandard Error t Stat P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 1509.246976.485

4 1.545590.13304

8 -487.8913506.38

3

-487.89

13506.38

3Gross Loan Portfolio (millions) 0.077729

0.046783

1.661468

0.107394 -0.01795

0.173411

-0.0179

50.17341

1


Politics -52.319956.2823

2 -0.92960.36025

4 -167.4362.7904

2-

167.4362.7904

2

Prop -10.875110.6704

6-

1.019180.31654

7 -32.698610.9484

7

-32.698

610.9484

7

Educ 18.6270532.0718

20.58079

20.56586

7 -46.967284.2212

8

-46.967

284.2212

8

Corrupt -121.35170.3870

6-

1.724050.09534

6 -265.30822.6070

9

-265.30

822.6070

9

y= β 0+ β 1 x1+ β2 x2+ β3 x3+ β4 x 4:

predicted value of GDP=1509.246+.0777 (GLP)−52.3199 (pol)−10.8751(¿)+18.62705(educ)−121.351(corruption)

From this equation, we can see that a one dollar increase in microfinance loans, all else equal, will decrease GDP by approximately .0777%. This is a fairly insignificant number which may show that MFL’s don’t have too much effect on GDP. Because I used categorical variables for “Politics”

and “Corruption,” we need to interpret those slope coefficients differently. The following table summarizes “Politics:”

Intercept SlopeNo political

unrestβ0= 1509.246 β1+ β3+β4+ β5

Political unrest β0+ β2=1456.9261 β1+ β3+β4+ β5

Cross-border war

β0+2 β2=1404.6242 β1+ β3+β4+ β5

Civil war β0+3 β2=1352.2863 β1+ β3+β4+ β5


This table makes more sense than just looking at the intercept forβ2 because as a country becomes more politically unstable (like in the case

of war), they see a smaller increase in their level of GDP. Goodness-of-fit: R2=21.94 %

Adjusted R2=8.48 %

The x variables explain 21.94% of y without taking into account the number of degrees of freedom. A better look at how the variables explain y is the adjusted R2 because it takes into account the degrees of freedom (or number of variables we are testing). 8.48% of y is explained by all the

x’s. This seems like a small number, however taking into account how much goes into GDP, it’s fairly agreeable with economic theory.

To see if the regression had overall significance, I preformed an F-test using the critical value method and received the following results:

H o : β1=β2=β3=β4=β5=0

H A : At least one β1 , β2 , β3 , β4 , β5≠ 0

P−value=Significance of F=¿.1835Significance level=.05

Rejectionrule : Reject H 0if p−value< .05

Therefore, we fail to reject Ho and conclude that y is not statistically significant in its dependence on x. For individual significance of each slope coefficient, I preformed a T-test for n-small thereby using the t-table.

H 0 : β1=0

H A : β1 ≠ 0


T−stat :β1−0

S β1

= .0777.0468

=1.6603

Critical Value : t α2

,n−k−1=2.045

Rejection Rule : Reject H 0 if |t−stat|>Critical Value

Therefore, we fail to reject H 0 and conclude that β1 is not significantly different than zero at the 5% level. These results suggest that GDP is not significantly related to Microcredit Finance. Furthermore, by

looking at the regression output we can conclude that no|t−stat| is greater than the critical value. Therefore, none of the slope coefficients significantly explain y.

I also ran a regression including three interaction terms to see if that would change my goodness-of-fit. I specifically wanted to view the interaction among microfinance loans and 3

other variables: corruption, property rights, and politics. Below are the results:

Regression Statistics

Multiple R0.7070

21

R Square0.4998

79

Adjusted R Square0.3459

95

Standard Error263.57

7Observations 35

ANOVA

df SS MS FSignifica

nce F

Regression 8 180541622567

73.248

421 0.010662

Residual 26 180629369472

.81Total 34 3611709

Coeffici

entsStandard

Error t StatP-

valueLower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept -639.67 1105.29

-0.578

730.567

749 -2911.631632.2

88 -2911.631632.28

8Gross Loan Portfolio (millions)

6.029652 1.777333

3.392527

0.002226 2.376291

9.683013

2.376291

9.683013

Politics -53.764 58.24257 - 0.364 -173.483 65.955 -173.483 65.9553


0.9231 441 34 4

Prop5.4420

13 11.368390.478

6970.636

155 -17.92628.810

07 -17.92628.8100

7

Educ6.1464

14 28.481130.215

8070.830

823 -52.397464.690

22 -52.397464.6902

2

Corrupt71.200

19 79.926470.890

8210.381

195 -93.091235.49

14 -93.091235.491

4

Prop*GLP

-0.0480

2 0.015783

-3.042

360.005

309 -0.08046

-0.0155

7 -0.08046 -0.01557

Cor*GLP

-0.4583

6 0.127402

-3.597

750.001

322 -0.72024

-0.1964

8 -0.72024 -0.19648

Pol*GLP

-0.0319

8 0.039453

-0.810

510.425

004 -0.113070.0491

2 -0.11307 0.04912

Adding the interaction terms significantly affected the entire regression and the adjusted R2increased a significant amount to 34.6%. It also brought allover significance to the regression model because α > p−value , (.05>.0107¿, which means we reject H o and conclude that one or more of the independent variables significantly explain y. Furthermore, by analyzing the t-stat, we find that microfinance, or GLP, along with the two interaction terms: corruption and politics with GLP significantly explain variation in y. We conclude that the marginal effect of microfinance (GLP) on GDP is statistically different for countries with corrupt governments as well as those with strong property laws. Another way to check whether the marginal effect of GLP on GDP differs among countries with corruption, I checked the confidence interval (-.7202, -.1965) in which does not contain zero. Thereby, rejecting H 0 and concluding that the marginal effect of GLP on GDP does differ depending on the amount of corruption (which I already proved by looking at the t-stat).

Interpreting the interaction slope coefficient GLP:∂ GDP∂ GLP

= β1+ β6 property+ β7 corruption+ β8 political

¿6.0297−.0481 (¿ )−.4584 ( corr )−.0320 ( pol) To estimate the slope coefficient for a given amount of property laws, corruption, and politics, all I need to do is plug in the numbers into the above equation.For: prop=55, corr=8, and pol=0

6.0297−.0481 (55 )−.4584 (8 )−.0320(0)=-.28317 | G a l l a g h e r

To see the percentage change on GDP for each one unit increase of an independent variable, I constructed a log-linear regression by taking the

natural log of all the values of GDP and running the regression again. The equation I received was:

log GDP=6.0537+.0003 (GLP )−.328 ( pol )−.0288 (¿ )+.2353 (educ )−.3084 (corr)

A one unit increase in microfinance will increase GDP by .0003%, an increase in political unrest will decrease GDP by .328%, an increase in

property rights will also decrease GDP by .0288%, and increase in education will increase GDP by .2353%, and an increase in corruption

will decrease GDP by .3084%.The only slope coefficient which doesn’t make much sense is an increase

in property rights, it goes against economic theory.V. Conclusion

The results show us that there were not any statistically significant agents affecting GDP

between the variables: microfinance, corruption, education, political state, and property

rights. However, once I added in the interaction terms into the regression, I found that

two of those terms and microfinance did significantly impact y. This seems to hold with

economic theory because there are so many different variables affecting GDP such that

these variables also interact in a sometimes large degree with each other; which when

taken into account (by multiplying the two terms as an extra variable), this can have a

magnifying effect on y.


I mainly set out to determine whether microfinance has an effect on GDP. The results

declare that there is a definite positive correlation, however weak, and therefore is not

statistically significant to GDP. Using the log-linear model, I found that a one unit

increase in GLP will increase GDP by .0003%, all else held constant. Furthermore, using

the interaction terms, I found GLP to have a direct relationship with both politics and

corruption signifying that when working together, the two terms significantly affect GDP.

Subsequently, I feel that microfinance (GLP) alone does not have a great effect on GDP

but when working with other aspects of an economy, the converse is true.

Works Cited

Alchian, Albert, “Property Rights,” The Concise Encyclopedia of Economics, 2008, Liberty

Fund, Inc., (http://www.econlib.org/library/Enc/PropertyRights.html)

Boudreaux, Karol and Cowen, Tyler, Why Microcredit Works, Wilson Quarterly, January 9,

2008, (portfolio.com)

Harper, Malcom, Book Review “Why Microfinance Doesn’t Work,” Microfinance Focus, June 7,

2010 (http://www.microfinancefocus.com/news)

Khandker, Shahidur, Using Microcredit to Advance Women, The World Bank, November 1998,

(http://www1.worldbank.org/prem/PREMNotes/premnote8.pdf)


http://www1.worldbank.org/prem/PREMNotes/premnote8.pdf

http://www.microfinancefocus.com/news/2010/06/07/book-review-why-microfinance-doesn%E2%80%99t-work%E2%80%9D/

http://www.econlib.org/library/Enc/PropertyRights.html

econometrics of microfinance- multiple linear regression

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