24 - = a s , p3 efioun7L fkaf m e a n ( * l ~ p r&?e d;fG~c

for rna /~6 & ~ u l e S [u- d ; & c d , h I 7 .

7 A i s m o d e l spec ,o f ; cd ,o r c fur ther presume

a ) & C S , € , ) . See exavrg le 2.

- /f W e . - +A;* k A jc a K c t r fhe dage ra f r o c A, d Cr-dtr J;&r-e ~e ca r -

d c I n o f h e r ; f i + e r n 3 l u r , A &rwj

I - J one Xr vrnkler $ one - 6 GpAaler,

TAU, eve& +Adu fL.6 ~ r r r * ~ o r \ d i * 9 e s-+J .m a e r

be ;den*'& *herd a & p n d ' f i k r ,

6 6 f- r ~ h b s P n o f b e .

Th;s b,dlg-c +/,e P, ~ 0 0 * a v T f o r

Computer Problem 4: Seasonal i ty Study

Given below i s monthly data on V i s i t s t o National Parks (V), Personal Income

(Y), U. S. Populat ion (N), ,and the Consumer P r i ce Index (P), f o r the years 1971

through 1976.

1. From t h i s data you w i l l need t o const ruc t the fo l lowing time ser ies :

LO^ (v/N); LO^ (Y / (N+:P) ) ;

(Note t h a t t h i s ser ies w i l l run from July, 1971 through June, 1976).

Vt RCMAt = - (over the same time per iod)

CMAt - From RCMA, c a l c u l a t e by hand the seasonal adjustment factors, R C M . ;

1

i = Jan., Feb .,..., Dec.

Then const ruc t the Seasonally Adjusted V i s i t s ser ies ;

SAV, = V t/mt (from Ju ly , 1971 through June, 1976)

2 . Next run the f o l l o w i n g regressions:

(1 ) Log (V/N) = a, + a, ~ o g (Y/(P+H)) + El I I

(2) Log (v/N) = a. + a Log (Y/(PIN))+ E bi Si + i = I

1 i n month i; where Si {O otherwise

(3) Log (SAVIN) = a, + al Log (Y/(p*N)) +

Note: Run equations (1) and (2) twice; once over the e n t i r e sample, and

once over the abridged sample, so t h a t they may be compared w i t h

the t h i r d regression.

Compare and con t ras t the r e s u l t s (sSE, R ~ , ;,, t - r a t i o s , e t c . ) f o r these

th ree equations, and exp la in why the r e s u l t s vary. A

3. v I n a l l cases p l o t the Actual values (A) o v e r the F i t t e d values (#I, and

the res idua ls aga ins t time. ,'- - .

Lt . Compare the seasonal adjustment f a c t o r s computed by hand (RCMA; , I Jan.,

Feb .,..., Dec.) w i t h those impl ied by equation ( 2 ) , the dummy regression. Note A

t h a t you must c a l c u l a t e the l a t t e r (ebi = seasonal adjustment fac tor f o r month

i; f Jan., Feb .,..., Dec.).

n Jan Feb Mar Apr

June July Aug Sepl Oct

SAV Y 3583.4 3558.5 3364.2 4051.0 3788.8 4130.1

4075.04 4121.2 4088.75 4210.9 41 15.88 3977.5 4148.83 3884.2

Nov 1831 4191.38 3710.4 880.3 207800 122.8 0 0 0 0 0 0 0 0 Dsc 1248 4244.28 2458.8 801.3 207848 123.1 0 0 0 0 0 0 0 0 Jan 1273 4288.48 3807.7 902.4 208088 123.2 1 0 0 0 0 0 0 0 Feb Mar Apr May June July

Sepl Oct Nov 2055 4824.82 3948.7 984.1 208457 128.9 0 0 0 0 0 0 0 0 Dec 1718 4858.08 3386.2 882 200584 127.3 0 0 0 0 0 0 0 0 Jan 1858 4640.21 4053.3 1000.2 208711 127.7 1 0 0 0 0 0 0 0 Feb 1648 4803.20 4882.1 1012.8 208808 128.8 0 1 0 0 0 0 0 0 Mar 2252 4587.70 4485.8 1022.4 208805 128.8 0 0 1 0 0 0 0 0 &I 3358 4807.87 5210.8 1031.1 210034 130.7 0 0 0 1 0 0 0 0 May 4828 4827.38 4078.4 1037.8 210154 131.5 0 0 0 0 1 0 0 0 June 7818 4828.48 4878.5 1045.7 210288 132.4 0 0 0 0 0 1 0 0 July 10030 4802.83 4025.7 1054.1 210410 132.7 0 0 0 0 0 0 1 0 Au9 10288 4571.48 4423.2 1064 210558 135.1 0 0 0 0 0 0 0 1 Sepl 5818 4544.00 4487.3 1074.8 210715 135.5 0 0 0 0 0 0 0 0 Oct 4158 4512.13 4458.8 1088.2 210883 138.8 0 0 0 0 0 0 0 0 Nov 2258 4482.75 4334.8 1088.7 210884 137.8 0 0 0 0 0 0 0 0 Dsc 1483 4388.38 2848.2 1103.8 211087 138.5 0 0 0 0 0 0 0 0 Jan 1307 4358.08 3808.4 1104.2 211207 139.7 1 0 0 0 0 0 0 0 Feb 1448 4351.50 3828.0 1100.8 211311 141.5 0 1 0 0 0 0 0 0 Mar 1882 4357.00 3887.7 1117 211411 143.1 0 0 1 0 0 0 0 0 &I 2851 4388.87 4428.7 1127 211522 143.8 0 0 0 1 0 0 0 0 May 4148 4385.71 4275.2 1141.1 211837 145.5 0 0 0 0 1 0 0 0 June 8777 4307.88 4182.0 1152.8 211772 148.8 0 0 0 0 0 1 0 0 July 9832 4423.58 3848.3 1188.1 211801 148 0 0 0 0 0 0 1 0

j--- &I 10364 4450.08 4481.0 1173.8 212051 148.8 0 0 0 0 0 0 0 1 %PI 5880 4478.88 4522.4 1180.0 212218 151.7 0 0 0 0 0 0 0 0 Ocl 4418 4485.08 4738.7 1181.8 212383 153 0 0 0 0 0 0 0 0 NW 2381 4502.38 4575.1 1152.8 212518 154.3 0 0 0 0 0 0 0 0

Mar Apr

June July Aue Sepl Oct Nov Dsc Jan Feb Mar Apr M.Y June July Aue Sepl Oct N w Dsc

...... . . . . . . . ... .... .. ... . *-151. r . -u w: hr- , /=&A d.? ..6de.e~.. .- --

-. - - . . - .. V ; s ; + s an. B . . ~ , s o n a ~ , . . . . IhcocL.,. P e r r Lp :+& . . . . . . . . . . . . . . -.

s a,, & 4 & eel- --. - 4 , ; :L -Y - L.+ L.2 4.

V,. = 6~ j - - Am-

O r : Ccse a sex J u r n h q , 4

a Y ? / '\ d a r k , "dh ;r Ma/e f F;,,k 0 . ,

. - -- - -- - - - 71 -- - --om-= 9 I"" - 119 . f ( I c 1

-4 Y 9 9 ,bf% -- V v

1 --"y m "°.'ISU

--- -- -- - - -- d . -S !y. 230 W! % zrn

B

d Y L x = 4 0 + q , & ~ + b I S I + bLS, + -- - + b,, 5,,

8 ( , - e m Searorcal a are e I - I,.~., ~t

3 1, ,,,k&d $0 ,, -f-- ,, Leue !! ti -

_Yt = ' + P + Ei j y t = ) ~ ;C n o t

if ; ~ B ; I . , ~ r * ~ ~ I ; e s (0 L o r

;F not

Co,s;dc. r the f t M d rnomcid-5 o f E+ .

S i n ce St = A+ r) $ Et ;r linear in St,

- L s o k c *tP ""1 ;my3 I ~ m b d b ; I ; t i i s

, u t s ; A ~ +Le r a n p , (0, 1 )

T h e Pro~i- t Model tAe

d l , no change i the ikpl i , f~ons > 7

for godA- redrcrr,. Coqs;dler ah a fkcpfic%'ve

ar ruw$;oh t it W ~ L - 4eherd( .

&+ E t be i i d . ~ ( o ; ~ ' ) . - -

u Acre f' jx -6, 5t f i f i .dard N o r m a ( &.

I

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#

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- - -- -. -- .- .... ---. ... .... - . . . . . . . . . . .

r-. ! A k 0 u 3 ~ 2 G + 9 ) A&- -1-7 p~l3+-Z-d - -

-- . . . . .......... ... ... .... . h . -fl, .L~~...14 9- era L o 1 ) y- , .

i !

.... ;; . . - W1. ' tk . . . . . . . . . .

-. .. . -~ . -& - . . . . . . . . . . . b lnt in1 ?dd e . . . . . . . . - . . 9 -0.u -mi . 1? &A. - :a s ~?.fi. Je . ...... .'. Y . . . . . . . . . . . . f $ y l e L b- $-f- . . . a . . _ c 06% ... air1

B (k nzero ohs>~)... -- ..

...... . . . . . . . . . .

. . . . .......... -- .. -. . . ..-+ ~- ......... -.

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c.dnJ ; - / ; o~ . r~ \ dehs;+;es <-J; f : d ~ l ah 3; > 0 ,....).