Nominal and Real U.S. GDP

Nominal and Real U.S. GDP

1960-2001

Year

Problem Set #5-Key

Sonoma State University Dr. Cuellar

Economics 318- Managerial Economics

Use the data set for gross domestic product (gdp.xls) to answer the following questions.

(1) Show graphically nominal gdp for the years provided. Be sure to correctly label your

axes.

(2) Based on the data, calculate the average annual dollar growth of nominal gdp.

SUMMARY OUTPUT Nominal Simple Linear

Regression Statistics Multiple R 0.965225 R Square 0.931659 Adjusted R Square 0.92995 Standard Error 789.4347 Observations 42

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

Intercept -1288.68 248.0414 -5.19544 6.34E-06 -1789.99 -787.375 X Variable 1 234.6787 10.04977 23.35165 6.4E-25 214.3674 254.99

Based on the simple linear regression, nominal GDP grew approximately 234.68 billion dollars per year from 1960-2001.

(3) Based on your data, calculate the average annual growth rate of gdp, assume annual compounding.

The growth formula assuming annual compounding is,

Yt = Y0(1 + g)t

To find g, the average annual growth rate assuming annual compounding take the log of GDP and regress against time.

LogYt = LogY0 + Log(1 + g)t, where Y0 is GDP at time zero and g is the average annual

growth rate. SUMMARY OUTPUT

Nominal Growth Rate Compounded Annually

Regression Statistics Multiple R 0.994717 R Square 0.989462 Adjusted R Square 0.989198 Standard Error 0.043069 Observations 42 Coefficients Standard Error

t Stat P-value Lower 95% Upper 95%

Intercept 2.684734 0.013532 198.3931 1.7E-61 2.657384 2.712083 X Variable 1 0.033601 0.000548 61.28393 3.6E-41 0.032493 0.034709 Based on the summary output,

LogYt = Log(2.684734) + Log(0.033601)t

Taking the anti-log gives

Y_t = 483.8754(1.080441)t

(4) Based on your data, calculate the average annual growth rate of gdp, assume continuous compounding.

The growth formula assuming continuous compounding is, Yt = (Y0)egt

Which can be linearized by taking the natural logarithm of both sides to get,

LnY_t = LnY₀ + gt

SUMMARY OUTPUT

Nominal Growth Rate Continuous Compounding

Regression Statistics Multiple R 0.994717 R Square 0.989462 Adjusted R Square 0.989198 Standard Error 0.09917 Observations 42 Coefficients Standard

Error t Stat P-value

Lower 95% Upper 95% Intercept 6.181827 0.031159 198.3931 1.7E-61 6.118852 6.244803 X Variable 1 0.077369 0.001262 61.28393 3.6E-41 0.074818 0.079921 LnYt = Ln(6.181827) + .077369t

The average annual growth is 7.8%per year assuming continuous compounding. Exponentiating both sides

Yt = 483.8754(1.080441)t

(5) Based on your answer from (4), estimate gdp for the year 2005.

The prediction 2005 is period 2005-1959 = 46

Y_t = 483.8754e.077369*46 _{= $16,997.41}

Note that if no rounding is done, base 10 log and natural log methods result in the same answer,

Y_t = 483.8754(1.080441)46 _{= $16,997.41}

(6) Construct a 95% prediction interval for your estimate in (4).

$16,997.41 ± tdf,α/2 SE 1% 1 n% (x_p&¯x)2 j (xi&¯x) 2 $16,997.41 ± 2.02 (.09917) 1% 1 42% (46&21.5)2 (42&1)150.52

$16,997.41 ± .212

Nominal GDP in 2005 is expected to be between $16,997.618 and $16,997.194

(7) Calculate real gdp for the years provided.

(8) Graph real and nominal gdp together for the years provided. Be sure to correctly label

your axes.

(9) Based on the data, calculate the average annual dollar growth of real gdp.

SUMMARY OUTPUT Nominal Simple Linear

Regression Statistics Multiple R 0.987857 R Square 0.975861 Adjusted R Square 0.975258 S t a n d a r d Error 312.3749 Observations 42 Coefficients Standard Error

t Stat P-value Lower 95% Upper 95%

Intercept 1839.213 98.14859 18.73907 2.06E-21 1640.847 2037.579 X Variable 1 159.9117 3.976638 40.21278 5.72E-34 151.8746 167.9487

Based on the simple linear regression, nominal GDP grew on average approximately 159.9117 billion dollars per year from 1960-2001.

(10) Based on the data, calculate the average annual growth rate of real gdp, assume annual compounding.

SUMMARY OUTPUT

Real Growth Rate Compounded Annually

Regression Statistics Multiple R 0.996052 R Square 0.99212 Adjusted R Square 0.991923 S t a n d a r d Error 0.015192 Observations 42 Coefficients Standard

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

Intercept 3.396159 0.004773 711.4782 1.13E-83 3.386511 3.405806 X Variable 1 0.013725 0.000193 70.96707 1.07E-43 0.013334 0.014116 Based on the summary output,

LogYt = Log(3.396159) + Log(0.013725)t

Taking the anti-log gives

Yt = 2489.766(1.032108)t

The average annual growth rate of nominal GDP is 3.2% per year assuming annual compounding.

(11) Based on the data, calculate the average annual growth rate of real gdp, assume

continuous compounding. SUMMARY OUTPUT

Real Growth Rate Continuous Compounding

Regression Statistics Multiple R 0.996052 R Square 0.99212 Adjusted R Square 0.991923 S t a n d a r d Error 0.034981 Observations 42 Coefficients Standard

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

Intercept 7.819944 0.010991 711.4782 1.13E-83 7.79773 7.842158 X Variable 1 0.031603 0.000445 70.96707 1.07E-43 0.030703 0.032503

LnY_t = Ln(7.819944) + 0.031603 t

(12) Based on the answer from (11), estimate real gdp for the year 2005.

Yt = 2489.766e.031603*46 = $10,654

(13) Construct a 95% prediction interval for your estimate in (12).

$10,654 ± tdf,α/2 SE 1% 1 n% (x_p&¯x)2 j (xi&¯x) 2 $10,654 ± 2.02 (0.034981) 1% 1 42% (46&21.5)2 (42&1)150.52 $16,997.41 ± .075