is transformed into a linear model through the logarithms of the two terms above equality resulting linear function log yi

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1 MODEL FOR MACROECONOMIC - ANALYSE BASED ON THE REGRESSION FUNCTION PhD Professor Costat ANGHELACHE Uverstatea Artfe /Academa de Stud Ecoomce - Bucureşt PhD Professor Maro G.R. PAGLIACCI Uversta degl Stud d Peruga - Itala PhD Caddate Lga PRODAN Uverstatea Creştă Dmtre Catemr - Bucureşt Abstract Regresso fucto s the key to makg may mcro ad macroecoomc aalyss. After studyg logcal varables to be aalyzed, graphcal represetato of data seres ad prmary terpretato of formato we pass to fudamet the ecoometrc model to be used. Key words: gross domestc product, f al cosumpto, smple regresso, correlato *** The lear regresso model volves detfyg varables for defg the model ad specfyg the resdual varable, the cotet whch the regresso model s used. For the aalyss chroologcal seres (of tme) we use a temporal fucto whch, essetally, s also a regresso, wth a varable tme (t). The goal of usg the regresso model s to obta the parameters correspodg to the set of varables formulated by aalyzg depedece betwee varables, where data seres are recorded at the level of populato statstcs for a perod or a momet, ad to hghlght the depedece betwee varables a gve tme horzo. I theoretcal aalyss, the depedece betwee varables s stochastc. The cosderato of the resdual varable such a model s requred. Other factors that fluece outcome varable are grouped resdual varable. Sgle-factoral olear models are made lear by trasformatos that are appled to the varables regresso model. So, for eample, a model of b form y a s trasformed to a lear model through the logarthms of the two terms above equalty resultg lear fucto log y log a b log. Ths model s recommeded whe pots (log,log y ) are 1, located lke cloud of pots aroud a straght le. Sometmes, for estmato the parameter we usg other estmato techques, whch caot be made lear by elemetary trasformatos, estmato of parameter s doe by umercal methods. 18 Romaa Statstcal Revew r. 1 / 013

2 Lear regresso model s based o data seres for the two characterstcs. These are represeted by vectors (varable factor) ad y (varable outcome). Ths requres to defe the methods used to estmate the two parameters; specfy the methods used for testg propertes of estmators of regresso model ad establsh how to use the regresso model coductg predctos. I defg the lear regresso fucto most commoly four hypotheses are cosdered, amely: data seres are ot affected by regstrato errors. for each fed value of the characterstc factor, average resdual varable s zero, amely: X 0 E, for ay, the lack of correlato betwee resdues epresses that the resduals do ot show the pheomeo of covarace, whch mples hypothess of gap betwee resdual wth depedet varable, whch mples that cov( X, j ) 0,for ay j, showg a growth of factor varable values do ot automatcally lead to a crease of the resdual varable values. Based o four hypothess we defe lear regresso model by fucto: y b a, = 1,.., Hhypothess are made o the resdual varable, amely: E 0 0, j cov, j, j N0, y b a, = 1,.., Revsta Româă de Statstcă r. 1 /

3 Hhypothess are made o varable result: E y X b a 0, j covy, yj, j y N b a Whe betwee the two varables there s a lear depedece, usg data sets (y, ), 1,, the values of results varable s estmated by the relato yˆ aˆ, ad the resdues seres s estmate of equalty: e ˆ ( ˆ ˆ y y y b a ). Determato of the lear model parameters s usually made usg the method of least squares or mamum lkelhood. If we use the method of least squares the values of resultg characterstcs are estmated based o the relatoshp: yˆ aˆ, where â ad are estmators of the regresso parameters. Real values of the characterstc result are equal to the estmate obtaed usg the regresso model, adjusted by resdual error, respectvely: y yˆ e I estmato of parameters we start o the codto that the sum of squared dffereces betwee the actual ad epected to be mmal, achevg equalty: m aˆ, m e m yˆ a ˆ. aˆ, aˆ, aˆ, The optmal codtos of the fucto lead to a system of two equatos, respectvely: aˆ, yˆ a ˆ 0 aˆ, yˆ a ˆ 0 aˆ. Equatos are determed by applyg the method of momets. - The frst equato follows from the codto E defg equalty: 0 0 Romaa Statstcal Revew r. 1 / 013

4 Revsta Româă de Statstcă r. 1 / e 0 1 or e 0 - The secod equato of the system s determed startg by hypothess of msmatch of seres of the values of varable factor wth values of the resdual varable values ( 0, cov X ), satsfyg the equalty : e 0 1. I order to determe the two estmators we solve the lear system of results equatos: y a b y a b ˆ ˆ ˆ ˆ Testg f the soluto satsfes the codtos of secod order s doe by determg the secod order dervatves of the fucto 1. The resultg matr has two propertes: t s defed postve ad the matr determat s postve, respectvely: 0 ) ( X The calculato formulas of the two estmators, â ad results from solvg the lear system of equatos. Slope coeffcet s obtaed from the relatoshp: y y y y a ˆ 1. Bardse, G., Nymage, R., Jase, E. (005) The Ecoometrcs of Macroecoomc Modellg, Oford Uversty Press

5 Usg the least squares method has some dsadvatages, such as : - does ot provde acceptable results f the hypothess formulated are ot satsfed; - deotg by aˆ, the estmators calculated by the seres (, y ), 1, 1 ad by aˆ, b those evaluated for a seres of values (, y ), 1, 1, t follows that betwee the two pars of estmators there s o smple relatoshp recurrece; - estmators are dstorted f data seres have major chages. The use of mamum lkelhood estmato of the parameters s based o the specfcato of resdual dstrbuto fucto. Resdual varable has the property : 1 ˆ 1 0 e, f N ad from ths we get y Nb ~, a ~, ~. The regresso model s becomg specfed whe the parameters are solved a ~, b ~ ~ ad,. For lear regresso model, the lkelhood fucto s gve by: a~ b ~,, ~ f ( y / ). 1 Determg the form estmators s doe usg mamum codtos for log lkelhood fucto. L 1 a, b, l la, b, l( ) l 1 ( y b Based o the property of the logarthm fucto, we get: ~, ~ a~ ~, b, ~ L a~, b a ~ ma ~ ma ~ ~ ~ a, b, a, b, ~. We fd that the by the mamum lkelhood method we obta the same set of estmators for parameters model as the case of the least squares method. Whe we use mamum lkelhood method we obta drectly the estmator of resdual varable dsperso. The am of the smple regresso s to hghlght the relatoshp betwee a depedet varable eplaed (edogeous, result) ad a depedet varable (eplaatory factors, eogeous, predctors). Eemple: To buld a lear regresso model we defed the fal cosumpto as a depedet varable, whle GDP value was cosdered as a depedet varable. ) Romaa Statstcal Revew r. 1 / 013

6 To determe the parameters of the lear regresso model we cosdered a rage of data o the evoluto of the results of the two macroecoomc dcators. Year Evoluto of GDP ad fal cosumpto of Romaa durg GDP Y X FINAL CUNSUMPTION X mllo le Source: Romaa Statstcal Yearbook, Gross domestc product, by type of use, NIS, Bucharest, 008, 009, 010, 011, 01 I order to detfy the typology of the regresso fucto we performed a graphcal represetato of the pars of pots that cludes GDP values ad the correspodg fal cosumpto. Revsta Româă de Statstcă r. 1 / 013 3

7 Correlato GDP - fal cosumpto Based o the graphcal represetato we ca say that betwee GDP ad fal cosumpto, there s a drect ad lear form, respectvely Y a bx. where: Y s the depedet varable (eplaed, edogeous, result), a s the Y tercept (costat term), b s the slope, X s the vector of depedet varable (eplaatory, eogeous), s a varable, terpreted as a error (dsturbace, measuremet error). Based o the graph t s reasoable to assume that the average varable Y depeds o X through a lear relatoshp. The calculatos performed usg lear regresso model fucto we get parameters a = ad a = b = Cosequetly, regresso fucto becomes: Y X -7616, , X.. Based o the above data, usg Ecel / Data Aalyss, the followg results were obtaed: 4 Romaa Statstcal Revew r. 1 / 013

8 Regresso model estmato results Ecel Multple R s the multple correlato coeffcet, ths case, the smple correlato betwee ad y. We ote that betwee the GDP ad the value of fal cosumpto recorded Romaa durg there s a drect ad very strog lk, cocluso draw from the value of Multple R (0.994). R Square, R s the coeffcet of determato, whch shows the valdty of the chose model for eplag the varato of y; Multple R s obtaed from R Square: r=, ad ths eample s a value close to 1 dcatg that the model s chose correctly, the fal cosumpto,, epla varato gross product, y, at the rate of 98.85%. Adjusted R Square s a coeffcet of determato corrected wth degrees of freedom ad has the same meag as R. Stadard Error s the stadard error ad t shows how average observed values y devate from the theoretcal values that are o regresso straght, ŷ ( ths case to ± ). Ths value rased to the power s the dsperso resdues. Observatos s, the umber of observatos, here = 14. ANOVA s the aalyss of verso table. For the verso due to factor, Regresso, the resdual verso, due to other factors uregstered, Resdual, ad total verso, due to all factors, I total, we specfy: - df (degrees freedom), degrees of freedom: k - umber of eplaatory varables (the smple regresso k = 1), -k-1 for resdue (14-1-1= 1 degrees of freedom) ad -1 for total verso (14-1 = 13); Sum df for Regresso ad Resdual s equal to the Total df: k + ( - k - 1) = - 1. Revsta Româă de Statstcă r. 1 / 013 5

9 - SS stads for Sum Square, whch s the sum of squares of devatos, called versos, as follows: - Regresso:, - Resdual:, - Total:. Betwee these versos there s the relatoshp: Total = Regresso + Resdual, lke:. MS, short for Modfed Sum, called modfed amouts, actually, modfed dsperso: -Regresso: model chose, -Resdual:, the verso due to the regresso, dsperso of resdue. F, Fsher test of overall sgfcace of the regresso s the rato of the two dspersos corrected by degrees of freedom. Itercept s the ame for the costat term (costat) of model. Coeffcets - cotas estmates of the coeffcets a ad b. From the values show t follows that the estmated model the eample s: GDP = CF. Also, the valdty of ths model s cofrmed by regresso tests F-statstc values ( value table level hgher tha what s cosdered to be the bechmark aalyss valdty of ecoometrc models) ad the degree of the eutral rsk (reflected by the test Sgfcace F value). Lower 95%, Upper 95% - lower ad upper lmts of the cofdece terval for the parameter. The 0.05 threshold lmts are calculated automatcally, regardless of the talzato procedure Regresso. It ca therefore be terpreted as the populato lear model parameters are cluded the followg rages: <a < ; <b < Romaa Statstcal Revew r. 1 / 013

Predcted y - predcted y value for that observato, t s obtaed by replacg the X s observato the model estmated Y X -7616,88095 + 1,018078X.

10 Predcted y - predcted y value for that observato, t s obtaed by replacg the X s observato the model estmated Y X -7616, ,018078X. Note that the sum of adjusted values s equals wth the sum of the emprcal Y X whch allows us to say that the estmate parameters of regresso equato s correct. Resduals the value of the predcto error (dfferece betwee the observed ad predcted value). Rezduals stadard - stadard error value. It s obtaed by dvdg the resdue to stadard devato of resdue. The qualty aalyss of the aalyss model that we choose s facltated by the graph below: Revsta Româă de Statstcă r. 1 / 013 7

11 The depedet varable vs. resdue dagram The graph leads to the cocluso, by the shape of the cloud of pots, that there s o correlato betwee depedet varables ad resdue, that we ca say that the model s well chose. To aalyze the correlato betwee the evoluto of GDP ad the fal cosumpto we have researched the data seres that cludes the values of the two dcators from 1998 to 011. These were processed usg the software package Evews7. A frst step ths research was the detfcato of the type of ecoometrc model that reflects the evoluto of the pheomeo studed. To ths ed, we geerated pars of pots chart GDP CF. Dagram fal cosumpto vs. GDP 8 Romaa Statstcal Revew r. 1 / 013

12 As you ca see from the chart above, pars of pots follow a straght path, so t s possble to aalyze the pheomeo vestgated usg smple lear regresso model. I ths research we used GDP as depedet varable, whle the fal cosumpto s the depedet varable. Also, the model, I etered the free term C. Results obtaed usg Evews program s as follows: The regresso model s characterstcs From the aalyss results that the smple lear regresso model reflects the correlato betwee the value of GDP ad of fal cosumpto s as follows: GDP = CF. Coclusos As we ca see, the fal cosumpto s a etremely mportat factor for the evoluto of GDP. Thus, for a crease by a moetary ut of fal cosumpto we wll get a crease of 1. moetary uts of GDP. Also, t s oted that the value of the free term C s very hgh, whch allows us to state that the factors that were ot cosdered buldg the desg have a sgfcat fluece o the evoluto of gross domestc product. Negatve value of the free term shows that the varables whch were ot cluded the ecoometrc model have a egatve effect o the evoluto of GDP. Revsta Româă de Statstcă r. 1 / 013 9

13 Selectve bblography: - Aghelache C., (008) Tratat de statstcă teoretcă ş ecoomcă, Edtura Ecoomca, Bucureşt - Aghelache, C (coordoator) ş alţ (01) Modele statstco ecoometrce de aalză ecoomc utlzarea modelelor î studul ecoome Româe, Revsta Româă de Statstcă (Suplmet), ISBN X - Aghelache, C. ş alţ (01) Ecoometre, Edtura Artfe, Bucureşt - Aghelache, C. ş alţ (01) Elemete de ecoometre teoretcă ş aplcată, Edtura Artfe, Bucureşt - Aghelache, C., Mtruţ, C. (coordoator), Bugudu, E., Deatcu, C. (009) Ecoometre: stud teoretce ş practce, Edtura Artfe, Bucureşt - Bardse, G., Nymage, R., Jase, E. (005) The Ecoometrcs of Macroecoomc Modellg, Oford Uversty Press - Turdea, M.S., Proda L., (01) Statstcă petru afacer, Edtura ProUverstara, Bucurest, ISBN Voeagu, V., Ţţa, E. ş colectv (007) Teore ş practcă ecoometrcă, Edtura Meteor Press - *** Isttutul Naţoal de Statstcă Auarul Statstc al Româe, Edţle 008, 009, 010, 011, Romaa Statstcal Revew r. 1 / 013

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from