functional form (lecture handout)

8/18/2019 Functional Form (Lecture Handout)

1/21

Handout 6: Functional Form

In which you learn how to use OLS to model economic events that may be non-linear. In so doing you learn how to estimate the economic concept of elasticities (of demand, income etc) and how to test for appropriate functional form of your

model

So far considered models written in linear formY = b0 + b1X + u (1)

Implies a straight line relationship between y and X

Sometimes economic theory and/or observation of data will suggest that the relationshipbetween variables is non linear


2/21

2

One way to model a non-linear relationship is the equation

Y = a + b/X + e (2)

(where the line asymptotes to the value “a” as X ↑ - from below if b0)

However this is not a linear equation, unlike (1), since it does not trace out a straight linebetween Y and X and OLS only works (ie minimise RSS) if can somehow make (2) linear.

- The solution is to use algebra to transform equations like (2) so appear like (1)

In the above example do this by creating a variable equal to the reciprocal of X, 1/X, sothat the relationship between y and 1/X is linear (ie a straight line)

Y = a + b*(1/X) + e (3)

(3) is now linear in parameters

The only thing now need to be careful about is how to interpret the coefficients from thisspecification, since now

dY/d((1/X) = b but dY/dX = -b/X2

Log Linear Models

A useful functional form isY = b0Xb1exp(u)

To male this model linear in parameters take (natural) logs so that

LnY = Lnb0 + b1LnX + u (4)This is a useful specification because the estimated coefficients can be interpreted aselasticities

Since dLnY/dY = 1/Y then dLnY = dY/Y

which is the % change in y÷

100

Similarly

dLnX=dX/X is the % change in X ÷ 100

From (4)

dLnY/dLnX = b1 = (dY/Y)/(dX/X)

so b1 = % Δ in Y/ % Δ in X

= elasticity of y wrt X


3/21

3


4/21

4

Example. Using the food.dta file (posted on the course web site)

use f ood. dt a

r eg f ood gr i nc

Source | SS df MS Number of obs = 200- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 198) = 60. 29

Model | 50055. 4304 1 50055. 4304 Pr ob > F = 0. 0000Resi dual | 164391. 019 198 830. 257671 R- squar ed = 0. 2334

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 2295 Tot al | 214446. 449 199 1077. 62035 Root MSE = 28. 814

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -f ood | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -gr i ncno | . 0171574 . 0022097 7. 76 0. 000 . 0127999 . 021515

_cons | 57. 59873 3. 00802 19. 15 0. 000 51. 66686 63. 5306- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

predict fhat /* will give predicted (fitted) values for this model */

Now t r y Food = a + b( 1/ I ncome) + u

g onei nc=1/ gr i nc

r eg f ood onei nc


Model | 63337. 7437 1 63337. 7437 Pr ob > F = 0. 0000Resi dual | 151108. 706 198 763. 175281 R- squar ed = 0. 2954

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 2918

Tot al | 214446. 449 199 1077. 62035 Root MSE = 27. 626- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

f ood | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

onei nc | - 12440. 58 1365. 594 - 9. 11 0. 000 - 15133. 56 - 9747. 606 _cons | 97. 26136 3. 147255 30. 90 0. 000 91. 05492 103. 4678

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

predict fhat2 /* will give predicted (fitted) values for this model */

New model is a better fit (compare the R2 )

Coefficient now says if income increases by £1, food expenditure changes bydFood/dIncome = -b/Income2

(just differentiate the Food eqn.

Note that a non-linear effect is not constant – unlike a straight line – so slope effect willchange as value of income changes


5/21

5


6/21

6

Often a graph is useful to show the fitted lines from different models

t wo ( scat t er f ood gr i nc, yt i t l e( f ood) ) ( l i ne f hat gr i nc, cl pat t ern( l i ne) ) ( l i nef hat 2 gr i nc, cl pat t er n( dash) )

0

5 0

1 0 0

1 5 0

2 0 0

f o o d

0 2000 4000 6000 8000gross normal weekly household income

weekly household food expenditure Fitted values

Fitted values

Now try a logarthimic model

g l f =l og( f ood)g l i =l og( gr i ncno)

r eg l f l i


Model | 16. 7099146 1 16. 7099146 Pr ob > F = 0. 0000Resi dual | 29. 0921232 198 . 146929915 R- squar ed = 0. 3648

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 3616 Tot al | 45. 8020378 199 . 230160994 Root MSE = . 38331

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l f | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt erval ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l i | . 3725856 . 0349377 10. 66 0. 000 . 3036879 . 4414833 _cons | 1. 747382 . 2323849 7. 52 0. 000 1. 289115 2. 205649

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

pr edi ct l hat

g ehat =exp( l hat ) / * t hi s wi l l change a l og val ue i nt o a l evel */


7/21

7


8/21

8

t wo (scat t er f ood gr i nc, yt i t l e( f ood) ) ( l i ne f hat gr i nc, cl pat t ern( dash) ) ( l i neehat gr i nc)

0

5 0

1 0 0

1 5 0

2 0 0

f o o d

0 2000 4000 6000 8000gross normal weekly household income

weekly household food expenditure Fitted values

ehat

Semi-Log Models

Another common functional form is the semi-log model(log-lin model) in which the dependent variable is measured in logs and the X variables inlevels

X y 1exp0

β β =

Taking (natural) logs gives

LogY = Logβ0 + β1Xlog(exp)

which since log(exp) = 1 gives

LogY = Logβ0 + β1X

The interpretation of the estimated coefficient β1 is

X

Y dy

dX

ydLog== 1

)( β

= % change in y /100 w.r.t. unit change in X

This is called a semi-elasticity


9/21

9

So if, for example, wage and age are related by

Agewage Log 050.05.3)(^

+=

then the % change in wages following a unit increase in age (ie 1 year)

= (0.050*1)*100 = 0.05 = 5%

Also useful for variables like the level of GDP which looks like this over time

0

5 0 0 0 0 0

1 0 0 0 0 0 0

g d p

1950 1960 1970 1980 1990 2000year

The semi-log model:

Log(GDP) = a + bYear + u

Implies that the coefficient b gives the (constant) growth rate over the period

Using the data set gdp.dta (posted on the course web site)

use gdp. dta

g l gdp=l og( gdp) / * cr eat e l og of gdp */

Est i mat e semi - l og model usi ng OLS

r eg l gdp year



- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 9828 Tot al | 125. 921878 55 2. 2894887 Root MSE = . 19825

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l gdp | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -year | . 0919893 . 001639 56. 12 0. 000 . 0887033 . 0952754

_cons | - 170. 0515 3. 238013 - 52. 52 0. 000 - 176. 5433 - 163. 5597- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

OLS estimate on year variable says that (nominal) GDP in the UK has been growing byaround 9% a year


10/21

10


11/21

11

Testing Functional FormIf want to compare goodness of fit of models in which the dependent variable is in logs orlevels then cant just look at the R2. The TSS in Y is not the same as the TSS in LnY, socomparing R2 is not valid. The basic idea behind testing for the appropriate functional formof the dependent variable is to transform the data so as to make the RSS comparable

Do this by1. dividing each observation by the geometric mean

where geometric (rather than arithmetic) mean

= (y1*y2*…yn)1/n = exp1/nLn(y1*

y2…y

n)

2. rescale each y observation by dividing by this value

yi* = yi /geometric mean

3. regress y* (rather than y) on X, save RSSregress Lny* (rather than Lny) on X, save RSS

the model with the lowest RSS is the one with the better fit

More formally

BoxCox = N/2*log(RSSlargest/RSSsmallest) ~ χ2(1)

If estimated value exceeds critical value (from tables Chi-squared at 5% level with 1degree of freedom is 3.84) reject the null hypothesis that the models are the same(ie there is a significantly different in terms of goodness of fit).

Example (Box-Cox Test)

. u boxcox / * r ead i n dat a */

The dat a cont ai ns i nf o on GDP and empl oyment growt h f or 21 count r i es

. su empl gdpVar i abl e | Obs Mean Std. Dev. Mi n Max

- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

empl | 21 1. 108095 . 8418647 . 02 3. 02gdp | 21 3. 059524 1. 625172 1. 15 7. 73

The dat a show t hat gdp and empl oyment growt h ar e measur ed i n percent age poi nt s,wi t h a maxi mumof 7. 73 %poi nt annual GDP growt h and a mi ni mum1. 15% poi nts.


12/21

12


13/21

13

A l i near r egr essi on gi ves

. r eg empl gdpSource | SS df MS Number of obs = 21

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 19) = 26. 97Model | 8. 31618159 1 8. 31618159 Pr ob > F = 0. 0001

Resi dual | 5. 85854191 19 . 308344311 R- squar ed = 0. 5867- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 5649 Tot al | 14. 1747235 20 . 708736175 Root MSE = . 55529

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -empl | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -gdp | . 396778 . 0764018 5. 193 0. 000 . 2368672 . 5566888

_cons | - . 1058566 . 2632937 - 0. 402 0. 692 - . 6569367 . 4452235

Gdp i s measur ed i n percent age poi nt s, dempl / dgdp = βgdp and hence dempl = βgdp* dgdp so a 1 % point r i se i n gdp gr owt h rai ses empl oymentgr owt h by 0. 4 poi nt s a year

and a l og- l i n speci f i cat i on gi vesg l empl =l og( empl ) / * gener at e l og of dep. Var i abl e */

. r eg l empl gdpSource | SS df MS Number of obs = 21

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 19) = 5. 89Model | 6. 84252682 1 6. 84252682 Pr ob > F = 0. 0253

Resi dual | 22. 0706507 19 1. 1616132 R- squar ed = 0. 2367- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 1965

Tot al | 28. 9131775 20 1. 44565888 Root MSE = 1. 0778- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l empl | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

gdp | . 35991 . 1482915 2. 427 0. 025 . 0495322 . 6702877 _cons | - 1. 436343 . 5110381 - 2. 811 0. 011 - 2. 505958 - . 3667282

l og- l i n model so coef f i ci ent s ar e gr owt h r at es. Thi s t i me dl empl / dgdp = βgdp and hence dl empl = βgdp* dgdp wher e dl empl = % change i n gdp/ 100.So a 1% point ( not a 1 %) r i se i n gdp gr owt h rai ses emp gr owt h by 36% a year( f r om t abl e of means above, can see a 35% i ncr ease i n gdp amount s t o ar ound 0. 36per cent age poi nt s of extr a gr owt h a year – whi ch i s si mi l ar t o est i mat e i nl evel s )

Looks l i ke l i near speci f i cat i on i s pr ef er r ed, but s i nce R2 or RSS not compar abl euse Box- Cox t est t o test f ormal l y

Get geomet r i c mean. means empl

Vari abl e | Type Obs Mean [ 95% Conf . I nt erval ]- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

empl | Ar i t hmet i c 21 1. 108095 . 724883 1. 491307| Geomet r i c 21 . 7152021 . 413749 1. 236291

Rescal e l i near dependent var i abl e and l og of dependent var i abl e

. g empadj =empl / . 715

. g l empadj =l og( empadj )

Regr ess adj ust ed dependent var i abl es on gdp and l og( gdp) r espect i vel y


14/21

14


15/21

15

. r eg empadj gdp

Source | SS df MS Number of obs = 21- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 19) = 26. 97


- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 5649 Tot al | 27. 7269772 20 1. 38634886 Root MSE = . 77663

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -empadj | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -gdp | . 5549343 . 1068557 5. 193 0. 000 . 3312828 . 7785858

_cons | - . 1480511 . 368243 - 0. 402 0. 692 - . 9187925 . 6226903

. r eg l empadj gdp

Source | SS df MS Number of obs = 21- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 19) = 5. 89

Model | 6. 84252671 1 6. 84252671 Pr ob > F = 0. 0253Resi dual | 22. 0706501 19 1. 16161317 R- squar ed = 0. 2367- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 1965

Tot al | 28. 9131769 20 1. 44565884 Root MSE = 1. 0778

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l empadj | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -gdp | . 35991 . 1482915 2. 427 0. 025 . 0495322 . 6702877

_cons | - 1. 100871 . 5110381 - 2. 154 0. 044 - 2. 170486 - . 0312554

Now RSS are compar abl e, and can see l i near i s pref err ed.

For mal t est of si gni f i cant di f f er ence bet ween t he 2 speci f i cat i ons

. g t est =( 21/ 2) *l og( 22. 1/ 11. 5) = N/ 2l og( RSSl argest / RSSsmal l est) ~ χ2(1)

/ * st at a r ecogni ses “l og” as Ln or l oge */

. di t est

6. 86

Gi ven t est i s Chi - Squared wi t h 1 degree of f r eedom. Est i mated val ue exceedscri t i cal val ue ( f r om t abl es Chi - squar ed at 5% l evel wi t h 1 degr ee of f r eedom i s3. 84) so model s ar e si gni f i cant l y di f f er ent i n t er ms of goodness of f i t .


16/21

16


17/21

17

Test for Normality of Residuals

All the hypotheses, tests and confidence intervals done so far are based on theassumption that the (unknown true) residuals are normally distributed. If not then tests areinvalid

When choosing a functional form better to choose one which gives normally distributederrors

Do this by looking at the OLS residuals Since can show that if all Gauss-Markov assumptions are satisfied (see earlier notes) thenthe OLS residuals are also asymptotically normally distributed (ie approximately normal ifsample size is large)

A normal distribution should have following properties- symmetric about its mean (in the case of OLS residuals the mean will be zero)

A Non-symmetric distribution is said to be skewed. Can measure this by looking at the 3rd moment of the normal distribution relative to the 2nd (the mean is the 1st moment, the variance is the second moment)

ndmoment cubeof

rdmoment squareof

X E

X E Skewness

X

X

2

3

])([

])([32

23

=−

−=

µ

µ

Symmetry is represented by a value of zero for the skewness coefficient

Right skewness gives a value > 0 (more values clustered to close to left of mean and afew values a long way to the right of the mean tend to make the value >0)

Left skewness gives a value < 0

A distribution is said to display kurtosis if the height of the distribution is unusual (suggestsobservations more bunched or more spread out than should be). Measure this by

ndmoment squareof

thmoment

X E

X E Kurtosis

X

X

2

4

])([

)(22

4

=−

−=

µ

µ

A normal distribution should have a kurtosis value of 3

Can combine both these features to give the Jarque-Bera Test for Normality (inresiduals)

−+=

24

)3(

6*

22KurtosisSkewness

N JB

Can show that this is asymptotically Chi2 distributed with 2 degrees of freedom (1 for

skewness and 1 for kurtosis)


18/21

18


19/21

19

If estimated chi-squared > chi-squaredcritical

reject null that residuals are normally distributed

(If not suggests should try another functional form to try and make residuals normal,otherwise t stats may be invalid).

Exampl e: Jarque-Bera Test for Normality (in residuals)

. u wage / * r ead i n data */

1st r egr ess hour l y pay on year s of experi ence and get r esi dual s

. r eg hourpay xperSource | SS df MS Number of obs = 379

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 1, 377) = 7. 53Model | 136. 061219 1 136. 061219 Pr ob > F = 0. 0064

Resi dual | 6815. 41926 377 18. 0780352 R- squar ed = 0. 0196- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 0170

Tot al | 6951. 48048 378 18. 39016 Root MSE = 4. 2518- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -hour pay | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]

- - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -xper | . 0487259 . 017761 2. 743 0. 006 . 0138028 . 083649

_cons | 7. 26455 . 4333534 16. 764 0. 000 6. 412457 8. 116642- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

. pr edi ct res, r es i d

Check hi st ogr amof r esi dual s usi ng t he f ol l owi ng st ata command

. hi st r es, nor mal bi n( 50)/ * nor mal opt i on super i mposes a normal di st r i but i on on t he gr aph */

Resi dual s show si gns of r i ght skewness ( r esi dual s bunched t o l ef t – notsymmet r i c) and kur t osi s ( l ept okur t i c – si nce peak of di st r i but i on hi gher t hanexpect ed f or a nor mal di st r i but i on)

F r a c t i o n

Residuals-6.58027 20.4404

0

.073879


20/21

20


21/21

To t est mor e f or mal l y

. su res, det ai l

Resi dual s- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Per cent i l es Smal l est1% - 6. 253362 - 6. 5802685% - 4. 919813 - 6. 372607

10% - 4. 27017 - 6. 313276 Obs 37925% - 3. 011451 - 6. 253362 Sumof Wgt . 379

50% - . 9261839 Mean 1. 11e- 08Largest St d. Dev. 4. 246199

75% 1. 869452 16. 509790% 5. 383683 17. 73377 Var i ance 18. 0302195% 7. 480312 17. 9211 Skewness 1. 5055599% 16. 5097 20. 44043 Kurt osi s 6. 432967

Construct Jarque-Bera test

. jb = (379/6)*((1.50555^2)+(((6.43-3)^2)/4))

= 328.9

The statistic has a Chi2 distribution with 2 degrees of freedom, (one for skewness one forkurtosis).

From tables critical value at 5% level for 2 degrees of freedom is 5.99

So JB> χ 2critical, so reject null that residuals are normally distributed.

Suggests should try another functional form to try and make residuals normal, otherwise tstats may be invalid.

Remember this test is only valid asymptotically, so it relies on having a large sample size.Users with data sets smaller than 100 observations should be wary about using this test.

functional form (lecture handout)

Documents