Academia de Studii Economice

Facultatea de Relatii Economice Internationale

The relationship between GDP, Exports and Expenditures of the

US states

by xxxx

- 2011 –

Summary:

1) Introduction on the data:- States- Variables: GDP, Exports, Government

Expenditures2) Performing the simple linear regression analysis:

- Presenting the available figures- Starting the regression analysis- Doing the Scatter diagram

3) Hypothesis testing:- Steps in hypothesis testing- Measures of variation

4) Testing the homoscedasticity of the model5) The Durbin Watson test6) Multiple linear regression analysis7) Conclusion 8) Bibliography

1) Presenting the available data

In this paper I am going to study the relation between the states of the United States of America, according to GDP, Exports and Government Expenditures. Due to the fact that one of the states, Puerto Rico, does not have relevant data, I have chosen to not include in this paper. Furthermore, I shall present the USA and the variables taken into consideration.

The United States of America is a federal constitutional republic comprising fifty states and a federal district. The country is situated mostly in central North America, where its forty-eight contiguous states and Washington, D.C., the capital district, lie between the Pacific and Atlantic Oceans, bordered by Canada to the north and Mexico to the south. The state of Alaska is in the northwest of the continent, with Canada to the east and Russia to the west, across the Bering Strait. The state of Hawaii is an archipelago in the mid-Pacific. The country also possesses several territories in the Pacific and Caribbean.

The U.S. economy is the world's largest national economy, with an estimated 2011 GDP of $15.1 trillion (22% of nominal global GDP and over 19% of global GDP at purchasing-power parity). The country accounts for 41% of global military spending, and is a leading economic, political, and cultural force in the world.

The United States has a capitalist mixed economy, which is fueled by abundant natural resources, a well-developed infrastructure, and high productivity. According to the International Monetary Fund, the U.S. GDP of $15.1 trillion constitutes 22% of the gross world product at market exchange rates and over 19% of the gross world product at purchasing power parity (PPP).

The United States is the largest importer of goods and third largest exporter, though exports per capita are relatively low. In 2010, the total U.S. trade deficit was $634.9 billion. Canada, China, Mexico, Japan, and Germany are its top trading partners. In 2010, oil was the largest import commodity, while transportation equipment was the country's largest export. China is the largest foreign holder of U.S. public debt.

The economy of the United States is a wide subject, I will not present it thoroughly, I just wanted to give you an idea of how it works. Next, the variables chosen for this paper are going to be presented: GDP, exports and government expenditures.

First of all, GDP, Gross domestic product, refers to the market value of all final goods and services produced within a country in a given period. GDP per capita is often considered an indicator of a country's standard of living, although this can be problematic because GDP per capita is not a measure of personal income.

GDP per capita is not a measurement of the standard of living in an economy. However, it is often used as such an indicator, on the rationale that all citizens would benefit from their country's increased economic production. Similarly, GDP per capita is not a measure of personal income. GDP may increase while real incomes for the majority decline. The major advantage of GDP per capita as an indicator of standard of living is that it is measured frequently, widely, and consistently. It is measured frequently in that most countries provide information on GDP on a quarterly basis, allowing trends to be seen quickly. It is measured widely in that some measure of GDP is available for almost every country in the world, allowing inter-country comparisons. It

is measured consistently in that the technical definition of GDP is relatively consistent among countries.

The major disadvantage is that it is not a measure of standard of living. GDP is intended to be a measure of total national economic activity—a separate concept.

The argument for using GDP as a standard-of-living proxy is not that it is a good indicator of the absolute level of standard of living, but that living standards tend to move with per-capita GDP, so that changes in living standards are readily detected through changes in GDP. Also, it is widely used by economists to gauge economic recession and recovery.

Secondly, I will try to give you an introduction in exports. This term export is derived from the conceptual meaning as to ship the goods and services out of the port of a country. The seller of such goods and services is referred to as an "exporter" who is based in the country of export whereas the overseas based buyer is referred to as an "importer". In International Trade, "exports" refers to selling goods and services produced in home country to other markets. Any good or commodity, transported from one country to another country in a legitimate fashion, typically for use in trade. Export goods or services are provided to foreign consumers by domestic producers. Export of commercial quantities of goods normally requires involvement of the customs authorities in both the country of export and the country of import. The advent of small trades over the internet such as through Amazon and eBay has largely bypassed the involvement of Customs in many countries because of the low individual values of these trades. Nonetheless, these small exports are still subject to legal restrictions applied by the country of export.

Basic statistics on international trade normally do not record smuggled goods or international flows of illegal services. A small fraction of the smuggled goods and illegal services may nevertheless be included in official trade statistics through dummy shipments or dummy declarations that serve to conceal the illegal nature of the activities.

The final variable is government expenditures. It includes all government consumption, investment but excludes transfer payments made by a state. Government acquisition of goods and services for current use to directly satisfy individual or collective needs of the members of the community is classed as government final consumption expenditure. Government acquisition of goods and services intended to create future benefits, such as infrastructure investment or research spending, is classed as government investment (gross fixed capital formation), which usually is the largest part of the government gross capital formation.

Acquisition of goods and services is made through own production by the government (using the government's labour force, fixed assets and purchased goods and services for intermediate consumption) or through purchases of goods and services from market producers. Government expenditures that are not acquisition of goods and services, and instead just represent transfers of money, such as social security payments, are called transfer payments. Government spending can be financed by seigniorage, taxes, or government borrowing.

The first two types of government spending, namely government final consumption expenditure and government gross capital formation, together constitute one of the major components of gross domestic product.

Now that we are all accustomed to the variables selected for this paper, I am going to present the first steps made towards studying the relationship between them.

2) The regression analysis Presenting the available figures:

At the beginning I shall present the first table, with the variables and try to give a little explanation regarding them.

Table 1: Defining data

Nr crt StateGDP (million $) y1

Exports (million $)x1

Government Expenditures (million $) x2

1 Alabama 174,400 15,502 546742 Alaska 45,600 4,155 142153 Arizona 261,300 15,636 630294 Arkansas 105,800 5,219 273025 California 1,936,400 143,192 3459706 Colorado 259,700 6,727 478067 Connecticut 233,400 16,056 425898 Delaware 62,700 4,966 8137

9District of Columbia 104,700 1,501 49889

10 Florida 754,000 55,365 17568411 Georgia 404,600 28,950 8391712 Hawaii 68,900 684 2461013 Idaho 54,800 5,157 1489814 Illinois 644,200 50,058 11607015 Indiana 267,600 28,745 6114916 Iowa 147,200 10,880 2936917 Kansas 128,500 9,905 3470518 Kentucky 161,400 19,343 5001219 Louisiana 213,600 41,356 4835720 Maine 53,200 3,164 1424221 Maryland 300,000 10,163 92155

22Massachusetts 377,700 26,304 83890

23 Michigan 372,400 44,768 9200324 Minnesota 267,100 18,904 4569125 Mississippi 98,900 8,229 3284826 Missouri 246,700 12,926 6794227 Montana 37,200 1,389 1092528 Nebraska 89,600 5,820 1652629 Nevada 127,500 5,912 18894

30New Hampshire 61,600 4,367 11844

31 New Jersey 497,000 32,154 8064732 New Mexico 75,500 1,541 2747233 New York 1,114,000 69,696 194975

34North Carolina 407,400 24,905 84830

35North Dakota 33,400 2,536 8618

36 Ohio 483,400 41,494 10797537 Oklahoma 160,500 5,353 3751638 Oregon 168,900 17,671 3359439 Pennsylvania 575,600 34,928 13568740 Rhode Island 49,500 1,949 11517

41South Carolina 164,300 20,329 46904

42South Dakota 39,900 1,259 9499

43 Tennessee 250,300 25,943 6854644 Texas 1,306,432 206,961 22710845 Utah 116,900 13,809 2070246 Vermont 26,400 4,277 709247 Virginia 427,700 17,163 15555448 Washington 351,100 53,353 6656049 West Virginia 66,600 6,449 1980850 Wisconsin 251,400 19,790 6128051 Wyoming 38,200 983 6278

Total USA14,665,132.0

0 1,207,883.01 3,191,504.00Source: Regression analysis for US states Lucian Luta, Wikipedia, www.usgovernmentspending.com

The states are arranged in an alphabetical order, with GDP as the dependent variable and Exports and Government Expenditures as the independent variables. There is a strong connection between them, because exports and expenditures are the primarily components of GDP.

It is easily observed why the US is one of the great powers of the world (many still considering it the sole super power) by the big number in the GDP column, which is sustained

by a high amount on the exports side. Of course, such a big economy could exist only with an effort from the government, briefly explained by the expenditures.

For the ones interested in a different kind of ranking, it is clear that states like California, New York or Texas have the biggest individual GDP, the first and the last being also the leaders on exports and government expenditures. (although the last one is not really a matter of pride) Smaller states are at lower in this ranking. Wyoming, for example, is among the last ones in both GDP and exports, but with low government expenditures.

Starting the regression analysis:

In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value". The error of a sample is the deviation of the sample from the (unobservable) true function value, while the residual of a sample is the difference between the sample and the estimated function value.

In regression analysis, the distinction between errors and residuals is subtle and important, and leads to the concept of studentized residuals.Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the fitted function are the residuals.

However, because of the behavior of the process of regression, the distributions of residuals at different data points (of the input variable) may vary even if the errors themselves are identically distributed. Concretely, in a linear regression where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be higher than the variability of residuals at the ends of the domain: linear regressions fit endpoints better than the middle. This is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence.

Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of residuals, which is called studentizing. This is particularly important in the case of detecting outliers: a large residual may be expected in the middle of the domain, but considered an outlier at the end of the domain.

In regression analysis, an estimating equation is developed to describe the functional nature of the relationship that exists between the variables ':

The dependent variable (Y) is estimated from the independent variable (X) using a regression function. In this case I am going to estimated GDP

If we use the equation of a straight line to describe the relationship between the dependent and the independent variable we say that we use a simple linear regression model.

Y i=a+bxi+εb

b=n∑ xi y i−∑ x i∑ y in∑ xi

2−(∑ x i)2

a=∑ yi∑ x i

2−∑ x i∑ x i y iin∑ x i

2−(∑ x i)2

Yi=a+bx

Using the above theory, I have managed to do such a table,

Table 2: “Predicted and residual values”

xi^2 Xi*Yi PredictedResidual Yi-Ycaciula

240296776.3 2703463142 215957.7133 -41,55817260921.13 189450967.2 116676.2828 -71,076244476923.4 4085623525 217132.3473 44,16827234267.68 552132763.1 125986.0996 -20,186

20504020744 2.77277E+11 1333209.054 603,19145248582.06 1746925711 139181.125 120,519257809584.9 3747575418 220813.2626 12,58724656635.39 311339660.4 123771.5475 -61,0722251981.225 157119129.5 93454.965 11,2453065256733 41745029607 564747.8519 189,252

838076576.2 11712988846 333623.6012 70,976467918.2242 47130733.85 86309.85846 -17,41026593820.45 282599197.8 125446.0135 -70,6462505832772 32247552823 518318.038 125,882

826273702.8 7692155846 331833.636 -64,234118374979.9 1601539923 175521.2879 -28,32198113371.94 1272820697 166992.0503 -38,492374141495.3 3121917838 249567.1813 -88,1671710307987 8833613840 442174.7376 -228,575

10010842.47 168324349.9 108008.5581 -54,809103291997.4 3048980119 169249.8809 130,750

691879469 9934870410 310472.4187 67,2282004190608 16671673009 472031.3752 -99,631

357350833.6 5049185051 245725.9717 21,37467713989.13 813833366 152324.3867 -53,424167070095.5 3188735596 193419.0194 53,2811928704.203 51662539.85 92476.03216 -55,27633871808.47 521467446.6 131247.3002 -41,64734949526.44 753756087.4 132051.0732 -4,551

19073585.4 269027627.2 118537.3817 -56,9371033853606 15980336206 361657.8959 135,342

2374591.231 116343300.9 93807.66937 -18,3084857481529 77640937318 690138.214 423,862

620262208.4 10146323037 298235.6791 109,1646433468.729 84716706.59 102517.5897 -69,1181721711598 20057964050 443379.0651 40,021

28656650.03 859187097.7 127163.3189 33,337312266611.5 2984643228 234940.6231 -66,0411219943825 20104380808 385930.3466 189,670

3799172.032 96482751.16 97379.06966 -47,879413255553.8 3340003430 258193.9181 -93,8941586075.318 50249853.4 91343.97298 -51,444673024929.7 6493463823 307314.8584 -57,015

42832759323 2.7038E+11 1891162.274 -584,730190698883.2 1614316129 201152.1218 -84,25218296300.44 112923821.9 117750.6633 -91,351294579712.9 7340753951 230498.0465 197,2022846586682 18732383316 547149.2312 -196,049

41591927.34 429515412.1 136752.8629 -70,153391625192.3 4975085903 253476.3986 -2,076966855.1159 37561598.2 88928.12974 -50,728

91,929,781,927.30 901,377,711,358.32 14,665,132.00 0.00

Source: Regression analysis for US states Lucian Luta

After Microsoft Excel did the computations for me, I entered the formulas for a and b, giving me the linear regression model for GDP and Exports, as follows (see Note 1 at the end):

a= 80324.68897b= 8.749665967x average= 23,684

The regression equation is Y=80324.68897+8.749665967*Xi, which means that for each increase in one million dollars in exports, the GDP is expected to rise with 8.749665967 million dollars.

Having in mind the validity of the model, the residual values’ total is equal to 0 which means that the calculations are correct.

Scatter diagram

The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.

Diagram 1: “Distribution of Exports”

0 5,000,000 10,000,000 15,000,000 20,000,000

500,000

1,000,000

1,500,000Exports (million $)x1

Exports (million $)x1Linear (Exports (million $)x1)

Source: Regression analysis for US states Lucian Luta

As you can see, the scatter diagram shows that there is a linear relationship between the variables, with the slope moving upwards.

3) Hypothesis testing

In order to validate the linear regression, we have to run a test on the slope, to see if it is statistically significant.

Steps in hypothesis testing:

1. State the null and alternative hypothesis:

H0: β1=0; H1 β1 ≠0

2. Define α, level of significanceα=5%

3. T test

Computing the degrees of freedom (51-2=49)

4. Defining the acceptance or rejection area

Defining the acceptance and rejections regions requires the use of the level of significance and the degrees of freedom. I again used Microsoft Excel for computing TINV of 0.05 and 49, TINV(0.05, 49)=2.

This means that for values smaller than -2 or greater than 2, there are in the rejection area, whereas values between -2 and 2 are in the acceptance region.

Figure 1: Acceptance/rejection regions

Source: Econometrics notebook Lucian Luta

5. t=b1−β1sb1 Sb1= Syx

√∑ ( Xi−x )2 the formulas for standard error

syx=√∑ ( yi− y )2

n−k−1n−2

According to the computations made in Excel, Sb1= 0.64377067550090, which is also the number gave by the regression analysis, so that the validity of the computations is verified.

t= (8.7496-0)/0.6437= 13.59

6. Statistical decision

Due to the fact that t is greater than 2, the statistical decision is to reject the null hypothesis in favour of the alternative one.

7. Interpreting the results

According to the results, there is a linear relationship between GDP and exports, meaning that any growth in exports will determine a change in GDP. The validity is tested and verified by the regression made with Excel, as seen in the printout* bellow:

Next, we have the confidence interval estimate of the slope, as seen in the Excel printout below:

At 95% level of confidence, the confidence interval for the slope is (7.45, 10.04), which does not include 0. That means that there is a significant linear relationship between GDP and exports.

Measures of variation: the sum of squares – the sum of the squares of the deviation from the mean. There are 3 types of sums of squares: Error, Regression and Total sum of squares. I will try to briefly define each of them, according to this paper:

a) SSR – the regression sum of squares which measures the variation attributable to the relationship between GDP and Exports: SSR=∑ ( yi− y ) 2

b) SSE – the error sum of squares which measures the variation attributable to factors other than the relationship between GDP and Exports: SSE=∑ ( yi− y )2

c) SST – the total sum of squares which measures the variation of each of the US countries’ GDP according to the average GDP: SST=∑ ( yi− y)2

Furthermore, the sum of squares types will be exemplified in the Excel printout below:

df SSRegression 1 4.84774E+12Residual 49 1.28592E+12Total 50 6.13367E+12

Now, in order to check the validity of the model, we will test if there is at least one independent variable linearly related to the dependent variable.

Steps in hypothesis testing:

1. State the null and alternative hypothesis: H0: β1=β2=βm=0; H1: at least one parameter β is not equal to 0

2. Fischer testα=5%; k= no of coefficients=2F(α,k,n-k)=F(0.05, 2, 49) = 3.15

3. Determining the rejection and acceptance region The acceptance region is between -3.15 and 3.15, the other values being in the rejection area.Figure 2: Acceptance/rejection area

Source: Econometrics notebook Lucian Luta

4. Computing FcalcFcalc= MSR/MSE= 184.72, which means that the values are in the rejection region and the model is valid, due to the fact that 184.72>3.15.

The test shows that there is at least one independent variable linearly related to the dependent variable. Furthermore you have an Excel printout with Anova and the computations made by Excel:

ANOVA

Df SS MS FSignificance

F

Regression 1 4.84774E+12 4.84774E+12184.72282

7 3.02E-18

Residual 49 1.28592E+122624333896

9Total 50 6.13367E+12

The coefficient of determination - measures the proportion of variation in GDP that is explained by the independent variable, Exports, in the regression model. It is denominated by r2.

r2=SSR /SST

Given the sum of squares calculated earlier, the coefficient of determination: r2=0.79, which means that the exports explain 79% percent of the variability in GDP. The validity of the model is also shown by the Regression done by Excel.

Regression StatisticsMultiple R 0.889016288R Square 0.79034996Adjusted R Square 0.786071388Standard Error 161997.9598Observations 51

4) Testing the homoscedasticity of the model

Further on, we are testing the homoscedasticity of the model. In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance.

The variables chosen for the test are GDP, as dependent variable, and exports, as the independent one.

Steps to test the assumption:

1) Arrange the data from small to large, according to an independent variable

At the first step, as the theory states, I arranged the data in Excel, smallest to largest, according to exports. The printout from Excel shows exactly how that worked out:

Table 3: “US states by exports, smallest to largest”

GDP (million $) y1

Exports (million $) X1

12 Hawaii 68,900 684

51 Wyoming 38,200 983

42 South Dakota 39,900 1,25927 Montana 37,200 1,389

9District of Columbia 104,700 1,501

32 New Mexico 75,500 1,541

40 Rhode Island 49,500 1,949

35 North Dakota 33,400 2,53620 Maine 53,200 3,1642 Alaska 45,600 4,155

46 Vermont 26,400 4,277

30New Hampshire 61,600 4,367

8 Delaware 62,700 4,96613 Idaho 54,800 5,1574 Arkansas 105,800 5,219

37 Oklahoma 160,500 5,353

28 Nebraska 89,600 5,82029 Nevada 127,500 5,912

49 West Virginia 66,600 6,4496 Colorado 259,700 6,727

25 Mississippi 98,900 8,22917 Kansas 128,500 9,905

21 Maryland 300,000 10,16316 Iowa 147,200 10,880

26 Missouri 246,700 12,92645 Utah 116,900 13,8091 Alabama 174,400 15,5023 Arizona 261,300 15,636

7 Connecticut 233,400 16,05647 Virginia 427,700 17,163

38 Oregon 168,900 17,671

24 Minnesota 267,100 18,90418 Kentucky 161,400 19,343

50 Wisconsin 251,400 19,790

41South Carolina 164,300 20,329

34North Carolina 407,400 24,905

43 Tennessee 250,300 25,943

22Massachusetts 377,700 26,304

15 Indiana 267,600 28,745

11 Georgia 404,600 28,950

31 New Jersey 497,000 32,154

39 Pennsylvania 575,600 34,928

19 Louisiana 213,600 41,356

36 Ohio 483,400 41,494

23 Michigan 372,400 44,768

14 Illinois 644,200 50,058

48 Washington 351,100 53,35310 Florida 754,000 55,365

33 New York 1,114,000 69,6965 California 1,936,400 143,192

44 Texas 1,306,432 206,961Total USA 14,665,132 1207883.009752

Source: Regression analysis for US states Lucian Luta

2) Divide a data set in 3 parts (2/5, 1/5, 2/5)For the next part, I have divided the data into 3 sets: 21, 9 and 21, as follows:

Table 4: The first set of divided data

1 Hawaii 68,900 684

2 Wyoming 38,200 983

3South Dakota 39,900 1,259

4 Montana 37,200 1,389

5District of Columbia 104,700 1,501

6New Mexico 75,500 1,541

7Rhode Island 49,500 1,949

8North Dakota 33,400 2,536

9 Maine 53,200 3,16410 Alaska 45,600 4,15511 Vermont 26,400 4,277

12New Hampshire 61,600 4,367

13 Delaware 62,700 4,96614 Idaho 54,800 5,15715 Arkansas 105,800 5,219

16 Oklahoma 160,500 5,353

17 Nebraska 89,600 5,82018 Nevada 127,500 5,912

19West Virginia 66,600 6,449

20 Colorado 259,700 6,727

21 Mississippi 98,900 8,229Source: Regression analysis for US states Lucian Luta

and

Table 5: “The second set of divided data”

22 Kansas 128,500 9,905

23 Maryland 300,000 10,16324 Iowa 147,200 10,88025 Missouri 246,700 12,92626 Utah 116,900 13,809

27 Alabama 174,400 15,50228 Arizona 261,300 15,636

29 Connecticut 233,400 16,056

30 Virginia 427,700 17,163Source: Regression analysis for US states Lucian Luta

and

Table 6: “The third set of divided data”

31 Oregon 168,900 17,671

32 Minnesota 267,100 18,904

33 Kentucky 161,400 19,343

34 Wisconsin 251,400 19,790

35 South Carolina 164,300 20,329

36 North Carolina 407,400 24,905

37 Tennessee 250,300 25,943

38 Massachusetts 377,700 26,304

39 Indiana 267,600 28,74540 Georgia 404,600 28,950

41 New Jersey 497,000 32,154

42 Pennsylvania 575,600 34,928

43 Louisiana 213,600 41,35644 Ohio 483,400 41,494

45 Michigan 372,400 44,76846 Illinois 644,200 50,058

47 Washington 351,100 53,35348 Florida 754,000 55,365

49 New York 1,114,000 69,69650 California 1,936,400 143,19251 Texas 1,306,432 206,961

Source: Regression analysis for US states Lucian Luta

3) Do a regression analysis for the first and the last table – with the help of Excel, I have the selected regression analysis:

ANOVA

Df SS MS F Significance F

Regression 1 2.77667E+12 2.77667E+12 47.22096163 1.4825E-06

Residual 19 1.11723E+125880167222

2Total 20 3.8939E+12

and

ANOVADf SS MS F Significance F

Regression 1 14698350057 14698350057 6.540855829 0.019249803Residual 19 42696041372 2247160072Total 20 57394391429

4) Calculating the ratio for the 2 regression analysis of the SSE2/SSE1

SSE2/SSE1 = 26.1671044

5) Calculating the degrees of freedom, according to the formula: d.f.=(n-d-2k)/2The number of degrees of freedom for this example is df=(51-9-4)2=19.FINV(0.05, 19,19)= 2.168

The ratio of the two regression analysis is higher than 2.16, which means that the data exhibits heteroscedasticity, meaning there are sub-populations that have different variabilities than others.

5) The Durbin Watson test

The final test we are going to conduct in this paper is the Durbin Watson test - a number that tests for autocorrelation in the residuals from a statistical regression analysis. The Durbin-Watson statistic is always between 0 and 4. A value of 2 means that there is no autocorrelation

in the sample. Values approaching 0 indicate positive autocorrelation and values toward 4 indicate negative autocorrelation.

The Durbin Watson test is computed with the following formula:

D=∑i=1

33

(et−e t−1)2

∑i=1

33

et2

First of all, I made a table in Excel with the values needed to conduct the Durbin Watson test:

Table 7: “Variables needed for Durbin Watson test”

Nr crtResidual Yi-Ycaciula e i-1 (ei- E I -1)^2 ei^2

1 -41557.71330 1727043534.37159

2 -71076.28283-

41557.71330 871345947.13625 5051837980.41916

3 44167.65274-

71076.28283 13281164684.05030 1950781548.260944 -20186.09962 44167.65274 4141405442.04894 407478617.76639

5 603190.94580-

20186.09962388598940748.3330

0363839317090.1830

0

6 120518.87498603190.9458

0232972327948.6340

0 14524799225.72060

7 12586.73738120518.8749

8 11649346326.71100 158425957.820108 -61071.54752 12586.73738 5425542933.42421 3729733915.92773

9 11245.03500-

61071.54752 5229688106.53538 126450812.1242310 189252.14811 11245.03500 31686532317.16130 35816375563.16240

11 70976.39883189252.1481

1 13989152865.99190 5037649191.5198412 -17409.85846 70976.39883 7812130478.75510 303103171.64023

13 -70646.01347-

17409.85846 2834088200.63929 4990859219.89616

14 125881.96198-

70646.01347 38623245136.21850 15846268351.81600

15 -64233.63597125881.9619

8 36143940582.41380 4125959989.48932

16 -28321.28789-

64233.63597 1289696744.58602 802095347.58158

17 -38492.05030-

28321.28789 103444408.13352 1481637936.5739918 -88167.18129 - 2467618638.37835 7773451856.37739

38492.05030

19 -228574.73759-

88167.18129 19714281865.38240 52246410662.46910

20 -54808.55807

-228574.7375

9 30194685144.45540 3003978037.42908

21 130750.11910-

54808.55807 34432022673.63710 17095593645.73280

22 67227.58132130750.1191

0 4035112806.08239 4519547690.6116723 -99631.37517 67227.58132 27841911361.03780 9926410917.62550

24 21374.02834-

99631.37517 14642307678.89430 456849087.6652025 -53424.38670 21374.02834 5594802892.54244 2854165093.83914

26 53280.98057-

53424.38670 11386035403.57790 2838862890.6081327 -55276.03216 53280.98057 11784625013.31560 3055439731.47320

28 -41647.30025-

55276.03216 185742333.54058 1734497618.00247

29 -4551.07317-

41647.30025 1376130063.69062 20712266.9718630 -56937.38173 -4551.07317 2744325325.20549 3241865438.65149

31 135342.10410-

56937.38173 36971400672.90530 18317485142.60950

32 -18307.66937135342.1041

0 23608252888.94430 335170757.89402

33 423861.78599-

18307.66937195513827257.2510

0179658813623.2890

0

34 109164.32093423861.7859

9 99034494515.22850 11916848964.26370

35 -69117.58970109164.3209

3 31784439659.20080 4777241206.34976

36 40020.93492-

69117.58970 11911217555.79750 1601675231.4734737 33336.68107 40020.93492 44679249.43373 1111334304.9181838 -66040.62313 33336.68107 9875848590.61224 4361363903.45849

39 189669.65338-

66040.62313 65387745513.04810 35974577413.28630

40 -47879.06966189669.6533

8 56429395818.28770 2292405311.57906

41 -93893.91809-

47879.06966 2117366275.84045 8816067854.03393

42 -51443.97298-

93893.91809 1801997839.46491 2646482356.28278

43 -57014.85844-

51443.97298 31034764.73599 3250694082.54175

44 -584730.27369-

57014.85844278483559497.6000

0341909492971.2410

0

45 -84252.12180

-584730.2736

9250478380520.6570

0 7098420027.80071

46 -91350.66326-

84252.12180 50389290.87935 8344943678.29798

47 197201.95350-

91350.66326 83262612641.62460 38888610465.42740

48 -196049.23121197201.9535

0154646494276.7670

0 38435301057.30050

49 -70152.86290

-196049.2312

1 15849895553.57120 4921424172.84678

50 -2076.39862-

70152.86290 4634404988.78904 4311431.2249951 -50728.12974 -2076.39862 2366990941.13206 2573343146.98313

Total -0.00001 50728.129732285336022382.280

001285923609494.830

00Source: Regression analysis for US states Lucian Luta

With the help of Excel, I made the Durbin Watson test, and the result was: 1.77.

The result is close to 0, which means there is a positive autocorrelation between the variables. Positive serial correlation is serial correlation in which a positive error for one observation increases the chances of a positive error for another observation. With this, the simple regression analysis is over.

6) Multiple regression analysisWe are moving on to the multiple regression analysis, with the independent variables

Exports and Government Expenditures and the dependant variable GDP. Due to the fact that I have shown how the computations are made, step by step, the multiple regression analysis is a printout of Excel, as follows:

SUMMARY OUTPUT

Regression Statistics

Multiple R0.99936008

9

R Square0.99872058

8

Adjusted R Square0.99866836

7

Standard Error73849.6651

1Observations 52

ANOVA

df SS MS FSignificance

F

Regression 2 2.08606E+141.04303E+1

419124.9217

8 1.3236E-71Residual 49 2.67235E+11 5453773037Total 51 2.08873E+14

CoefficientsStandard

Error t Stat P-value Lower 95%Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

-1170.99271

4 10642.87282

-0.11002599

90.91283815

3-

22558.6459920216.6

6 -22558.6 20216.66

Exports (million $)x1 2.84264767 0.5196062785.47077237

7 1.51235E-06 1.7984597823.88683

6 1.79846 3.886836

Government Expenditures (million $) x23.52874130

4 0.19899291417.7329997

6 5.79953E-23 3.1288500783.92863

3 3.12885 3.928633

In order to check the validity of the model, we will test if there is at least one independent variable linearly related to the dependent variable.

Steps in hypothesis testing:

1. State the null and alternative hypothesis: H0: β1=β2=βm=0; H1: at least one parameter β is not equal to 0

2. Fischer testα=5%; k= no of coefficients=3

F(α,k,n-k)=F(0.05, 3, 49) = 2.793948865

3. Determining the rejection and acceptance region The acceptance region is between -2.79 and 2.79, the other values being in the rejection area.Figure 3: Acceptance/rejection region

Source: Econometrics notebook Lucian Luta

4. Computing Fcalc= MSR/MSE= 19124,92, which means that the values are in the rejection region and the model is valid, due to the fact that 19124,92 > 2.79.

The test shows that there is at least one independent variable linearly related to the dependent variable. Furthermore you have an Excel printout with Anova and the computations made by Excel:

ANOVA

Df SS MS FSignificance

F

Regression 2 2.08606E+14 1.04303E+1419124.9217

8 1.3236E-71Residual 49 2.67235E+11 5453773037Total 51 2.08873E+14

The coefficient of determination - measures the proportion of variation in GDP that is explained by the independent variable, Exports, in the regression model. It is denominated by r2.

r2=SSR /SST

Given the sum of squares calculated earlier, the coefficient of determination: r2=0.99, which means that the exports and government expenditures explain almost perfectly the variability in GDP. The validity of the model is also shown by the Regression done by Excel.

Regression StatisticsMultiple R 0.999360089R Square 0.998720588Adjusted R Square 0.998668367Standard Error 73849.66511Observations 52

*Note 1: all of the printouts from Excel are from “Regression analysis for US states Lucian Luta

7) Conclusion The goal of this paper was to show that there is a positive linear relationship between

GDP and two of its most important components: exports and government expenditures in the US states. I have performed simple regression analysis on GDP and exports, explaining it step by step and also multiple regression analysis, with the help of Microsoft Excel.

8) BibliographyRegression analysis for US states xxxxxStatistics for Economists, Erika Tusa

www.usgovernmentspending.com

www.wikipedia.com