4 Hypothesis testing in the multiple regression model

Transcription

1 4 Hypothess testng n the multple regresson model Ezequel Urel Unversdad de Valenca Verson: Hypothess testng: an overvew Formulaton of the null hypothess and the alternatve hypothess 4.1. Test statstc Decson rule 3 4. Testng hypotheses usng the t test Test of a sngle parameter Confdence ntervals Testng hypothess about a sngle lnear combnaton of the parameters Economc mportance versus statstcal sgnfcance Testng multple lnear restrctons usng the F test Excluson restrctons 4.3. Model sgnfcance Testng other lnear restrctons Relaton between F and t statstcs Testng wthout normalty Predcton Pont predcton Interval predcton Predctng y n a ln(y) model Forecast evaluaton and dynamc predcton 34 Exercses Hypothess testng: an overvew Before testng hypotheses n the multple regresson model, we are gong to offer a general overvew on hypothess testng. Hypothess testng allows us to carry out nferences about populaton parameters usng data from a sample. In order to test a hypothess n statstcs, we must perform the followng steps: 1) Formulate a null hypothess and an alternatve hypothess on populaton parameters. ) Buld a statstc to test the hypothess made. 3) Defne a decson rule to reect or not to reect the null hypothess. Next, we wll examne each one of these steps Formulaton of the null hypothess and the alternatve hypothess Before establshng how to formulate the null and alternatve hypothess, let us make the dstncton between smple hypotheses and composte hypotheses. The hypotheses that are made through one or more equaltes are called smple hypotheses. The hypotheses are called composte when they are formulated usng the operators "nequalty", "greater than" and "smaller than". It s very mportant to remark that hypothess testng s always about populaton parameters. Hypothess testng mples makng a decson, on the bass of sample data, on whether to reect that certan restrctons are satsfed by the basc assumed model. The restrctons we are gong to test are known as the null hypothess, denoted by H. Thus, null hypothess s a statement on populaton parameters. 1

2 Although t s possble to make composte null hypotheses, n the context of the regresson model the null hypothess s always a smple hypothess. That s to say, n order to formulate a null hypothess, whch shall be called H, we wll always use the operator equalty. Each equalty mples a restrcton on the parameters of the model. Let us look at a few examples of null hypotheses concernng the regresson model: a) H : 1 = b) H : 1 + = c) H : 1 = = d) H : + 3 =1 We wll also defne an alternatve hypothess, denoted by H 1, whch wll be our concluson f the expermental test ndcates that H s false. Although the alternatve hypotheses can be smple or composte, n the regresson model we wll always take a composte hypothess as an alternatve hypothess. Ths hypothess, whch shall be called H 1, s formulated usng the operator nequalty n most cases. Thus, for example, gven the H : H : 1 (4-1) we can formulate the followng H 1 : H : 1 1 (4-) whch s a two sde alternatve hypothess. The followng hypotheses are called one sde alternatve hypotheses H : 1 1 (4-3) H : 1 1 (4-4) 4.1. Test statstc A test statstc s a functon of a random sample, and s therefore a random varable. When we compute the statstc for a gven sample, we obtan an outcome of the test statstc. In order to perform a statstcal test we should know the dstrbuton of the test statstc under the null hypothess. Ths dstrbuton depends largely on the assumptons made n the model. If the specfcaton of the model ncludes the assumpton of normalty, then the approprate statstcal dstrbuton s the normal dstrbuton or any of the dstrbutons assocated wth t, such as the Ch-square, Student s t, or Snedecor s F. Table 4.1 shows some dstrbutons, whch are approprate n dfferent stuatons, under the assumpton of normalty of the dsturbances. TABLE 4.1. Some dstrbutons used n hypothess testng. Known 1 restrcton 1 or more restrctons N Ch-square Unknown Student s t Snedecor s F

3 The statstc used for the test s bult takng nto account the H and the sample data. In practce, as s always unknown, we wll use the dstrbutons t and F Decson rule We are gong to look at two approaches for hypothess testng: the classcal approach and an alternatve one based on p-values. But before seeng how to apply the decson rule, we shall examne the types of mstakes that can be made n testng hypothess. Types of errors n hypothess testng In hypothess testng, we can make two knds of errors: Type I error and Type II error. Type I error We can reect H when t s n fact true. Ths s called Type I error. Generally, we defne the sgnfcance level () of a test as the probablty of makng a Type I error. Symbolcally, Pr( Reect H H ) (4-5) In other words, the sgnfcance level s the probablty of reectng H gven that H s true. Hypothess testng rules are constructed makng the probablty of a Type I error farly small. Common values for are.1,.5 and.1, although sometmes.1 s also used. After we have made the decson of whether or not to reect H, we have ether decded correctly or we have made an error. We shall never know wth certanty whether an error was made. However, we can compute the probablty of makng ether a Type I error or a Type II error. Type II error We can fal to reect H when t s actually false. Ths s called Type II error. Pr( No reect H H ) (4-6) 1 In words, s the probablty of not reectng H gven that H 1 s true. It s not possble to mnmze both types of error smultaneously. In practce, what we do s select a low sgnfcance level. Classcal approach: Implementaton of the decson rule The classcal approach mples the followng steps: a) Choosng. Classcal hypothess testng requres that we ntally specfy a sgnfcance level for the test. When we specfy a value for, we are essentally quantfyng our tolerance for a Type I error. If =.5, then the researcher s wllng to falsely reect H 5% of the tme. b) Obtanng c, the crtcal value, usng statstcal tables. The value c s determned by. 3

4 The crtcal value (c) for a hypothess test s a threshold to whch the value of the test statstc n a sample s compared to determne whether or not the null hypothess s reected. c) Comparng the outcome of the test statstc, s, wth c, H s ether reected or not for a gven. The reecton regon (RR), delmted by the crtcal value(s), s a set of values of the test statstc for whch the null hypothess s reected. (See fgure 4.1). That s, the sample space for the test statstc s parttoned nto two regons; one regon (the reecton regon) wll lead us to reect the null hypothess H, whle the other wll lead us not to reect the null hypothess. Therefore, f the observed value of the test statstc S s n the crtcal regon, we conclude by reectng H ; f t s not n the reecton regon then we conclude by not reectng H or falng to reect H. Symbolcally, If s c reect H (4-7) If s c not reect H If the null hypothess s reected wth the evdence of the sample, ths s a strong concluson. However, the acceptance of the null hypothess s a weak concluson because we do not know what the probablty s of not reectng the null hypothess when t should be reected. That s to say, we do not know the probablty of makng a type II error. Therefore, nstead of usng the expresson of acceptng the null hypothess, t s more correct to say fal to reect the null hypothess, or not reect, snce what really happens s that we do not have enough emprcal evdence to reect the null hypothess. In the process of hypothess testng, the most subectve part s the a pror determnaton of the sgnfcance level. What crtera can be used to determne t? In general, ths s an arbtrary decson, though, as we have sad, the 1%, 5% and 1% levels for are the most used n practce. Sometmes the testng s made condtonal on several sgnfcance levels. Non Reecton Regon NRR Reecton Regon RR FIGURE 4.1. Hypothess testng: classcal approach. An alternatve approach: p-value Wth the use of computers, hypothess testng can be contemplated from a more ratonal perspectve. Computer programs typcally offer, together wth the test statstc, a probablty. Ths probablty, whch s called p-value (.e., probablty value), s also known as the crtcal or exact level of sgnfcance or the exact probablty of makng a c W 4

5 Type I error. More techncally, the p value s defned as the lowest sgnfcance level at whch a null hypothess can be reected. Once the p-value has been determned, we know that the null hypothess s reected for any p-value, whle the null hypothess s not reected when <p-value. Therefore, the p-value s an ndcator of the level of admssblty of the null hypothess: the hgher the p-value, the more confdence we can have n the null hypothess. The use of the p-value turns hypothess testng around. Thus, nstead of fxng a pror the sgnfcance level, the p-value s calculated to allow us to determne the sgnfcance levels of those n whch the null hypothess s reected. In the followng sectons, we wll see the use of p value n hypothess testng put nto practce. 4. Testng hypotheses usng the t test 4..1 Test of a sngle parameter The t test Under the CLM assumptons 1 through 9, If we typfy ˆ ~, var( ˆ N ) 1,,3,, k (4-8) ˆ ˆ ~ N,1 1,,3,, k (4-9) ˆ ( ˆ var( ) sd ) The clam for normalty s usually made on the bass of the Central Lmt Theorem (CLT), but ths s restrctve n some cases. That s to say, normalty cannot always be assumed. In any applcaton, whether normalty of u can be assumed s really an emprcal matter. It s often the case that usng a transformaton,.e. takng logs, yelds a dstrbuton that s closer to normalty, whch s easy to handle from a mathematcal pont of vew. Large samples wll allow us to drop normalty wthout affectng the results too much. Under the CLM assumptons 1 through 9, we obtan a Student s t dstrbuton bˆ -b t (4-1) n-k se( bˆ ) where k s the number of unknown parameters n the populaton model (k-1 slope parameters and the ntercept, 1 ). The expresson (4-1) s mportant because t allows us to test a hypothess on. If we compare (4-1) wth (4-9), we see that the Student s t dstrbuton derves from the fact that the parameter n sd( ˆ ) has been replaced by ts estmator ˆ, whch s a random varable. Thus, the degrees of freedom of t are n-1-k correspondng to the degrees of freedom used n the estmaton of ˆ. When the degrees of freedom (df) n the t dstrbuton are large, the t dstrbuton approaches the standard normal dstrbuton. In fgure 4., the densty functon for normal and t dstrbutons for dfferent df are represented. As can be seen, 5

6 the t densty functons are flatter (platycurtc) and the tals are wder than normal densty functon, but as df ncreases, t densty functons are closer to the normal densty. In fact, what happens s thatt the t dstrbuton takes nto account that s estmated because t s unknown. Gven ths uncertanty, the t dstrbuton extends more than the normal one. However, as the df grows the t-dstrbuton s nearer to the normal dstrbuton becausee the uncertanty of not knowng decreases. Therefore, the followng convergence n dstrbuton should be kept n mnd: tn N (,1) n (4-11)( Thus, when the number of degrees of freedom of a Student s S t tends to nfnty, the t dstrbuton converges towards a dstrbuton N(.1). In the context of testng a hypothess, f the sample sze grows, so wll the degrees of freedom. Ths means that for large szes the normal dstrbuton can be used to test hypothess wth one unque restrcton, even when you do not know the populaton varance. As a practcal rule, when the df are larger than 1, we can take the crtcal values from the normal dstrbuton. FIGURE E 4.. Densty functons: normal and t for dfferent degrees of freedom. Consder the null hypothess, H : Snce measures the partal effect of x on y after controllng for all other ndependent varables, H : means that, once x, x 3,,x 1, x +1,, x k have been accounted for, x has no effect on y. Ths s called a sgnfcance test. Thee statstc we use to test H :, aganstt any alternatve, s called the t statstc or the t rato of ˆ and s expressed as t ˆ ˆ se( ˆ ) In order to test H :, t s natural to look at our unbasedd estmator of, ˆ. In a gven sample ˆ wll never be exactly zero, but a small value wll ndcate that 6

7 the null hypothess could be true, whereas a large value wll ndcate a false null hypothess. The queston s: how far s ˆ from zero? We must recognze that there s a samplng error n our estmate ˆ, and thus the sze of ˆ must be weghted aganst ts samplng error. Ths s precsely what we do when we use t ˆ, snce ths statstc measures how many standard errors ˆ s away from zero. In order to determne a rule for reectng H, we need to decde on the relevant alternatve hypothess. There are three possbltes: one-tal alternatve hypotheses (rght and left tal), and two-tal alternatve hypothess. One-tal alternatve hypothess: rght Frst, let us consder the null hypothess aganst the alternatve hypothess H : H : 1 Ths s a postve sgnfcance test. In ths case, the decson rule s the followng: Decson rule If If t t reect H ˆ nk t t not reect H ˆ nk (4-1) Therefore, we reect H : n favor of H : 1 at when t ˆ t nk as can be seen n fgure 4.3. It s very clear that to reect H aganst H : 1, we must get a postve t ˆ. A negatve t ˆ, no matter how large, provdes no evdence n favor of H : 1. On the other hand, n order to obtan t n k n the t statstcal table, we only need the sgnfcance level and the degrees of freedom. It s mportant to remark that as decreases, t ncreases. n k To a certan extent, the classcal approach s somewhat arbtrary, snce we need to choose n advance, and eventually H s ether reected or not. In fgure 4.4, the alternatve approach s represented. As can be seen by observng the fgure, the determnaton of the p-value s the nverse operaton to fnd the value of the statstcal tables for a gven sgnfcance level. Once the p-value has been determned, we know that H s reected for any level of sgnfcance of >p-value, whle the null hypothess s not reected when <p-value. 7

8 FIG URE 4.3. Reecton regon usng t: rght-tal alternatve hypothess. EXAMPLE 4.1 Is the margnal propensty to consume smaller than the average a propensty to consume? As seen n example 1.1, testng the 3rd proposton of the Keynesan consumpton functon n a lnear model, s equvalent to testng whether the ntercept s sgnfcatve1y greater than. That s to say, n the model we must test whether 1 Wth a random sample of 4 observatons, the followng results have been obtaned The numbers n parentheses, below the estmates, are standard errors e (se) of the estmators.. The queston we posee s the followng: s the thrd proposton of thee Keynesan theory admssble? Next, we answer ths queston. 1) In ths case, the null and alternatve hypotheses are the followng: H : 1 H : ) The testt statstc s: conss nc u cons ˆ t se( ˆ ) ) Decson rule It s useful to use several sgnfcance levels. Let us begn wth w a sgnfcance level of.1 because the value of t s relatvely small (smaller than 1.5). In ths case, the t degrees off freedom are 4 (4 observatons mnus estmated parameters). p If we look at the t statstcal table (row 4 and column.1,.1 or., n statstcal tables wth one tal, or two tals, respectvely), we fnd t As t<1.33, we do not reect H for =.1, and therefore we cannot reect for = =.5 (.5 t ) or =.1 ( t ), as can been n fgure 4.5. In ths fgure, the reecton regon corresponds to =.1. Therefore, we cannott reect H n favor H 1. In other words, the sample data are not consstent wth Keynes s proposton 3. In the alternatve approach, as can be seen n fgure 4.6, thee p-value correspondng to a t ˆ 1 =1.171 for a t wth 4 df s equal to.14. For <.14 - for example,.1,.5 and.1-, H s not reected. FIGU 1 = nc (.35) (.6) 1 1 ˆ se( ˆ ).35 1 URE 4.4. p-value usng t: rght-tal alternatve hypothess. 8

9 FIGU URE 4.5. Example 4.1: Reecton regon usng t wth a rght-tal alternatve hypothess. GURE 4.6. Example 4.1: p-value usng t wth rght-tal alternatve hypothess. FIG One-taaganst the alternatve hypothess H : 1 alternatve hypothess: left Consder now the null hypothess H : Ths s a negatve sgnfcance test. In ths case, the decson rule ss the followng: Decson rule If I If I t ˆ t ˆ t t nk nk reect not reect H H (4-13)( t ˆ t n H : 1 prov Therefore, we reect H : n favor of H 1 : at a gven when, as can be seen n fgure, we must get a negatve 4.7. It s t ˆ des no evdence n favor of H : 1.. A postve very clearr that to reect H aganst In fgure 4.8 the alternatve approach s represented. Once the p-value has been determned, we know thatt H s reected for any level of sgnfcance of >p-value, whle the null hypothess ss not reected when <p-value. t ˆ, no matter how large t s, 9

10 Reecton Regon RR Non Reecton Regon NRR tn k Non reected for α>p-value Reected for ɑ<p-value tn k p-value GURE 4.7. Reecton regon usng t: left-tal alternatve hypothess. FIG t nk FIGU t ˆ URE 4.8. p-value usng t: left-tal alternatve hypothess. MPLE 4. Has ncome a negatve nfluence on nfant mortalty? The followng model has been used to explan the deaths of chldren under 5 years per 1 lve brths (deathun5). deathun5 gnpc ltrate u 1 3 EXAM wheree gnpc s the gross natonall ncome per capta and ltrate s the adult (% 15 and older) llteracy rate n percentage. Wth a sample of 13 countres (workfle hdr1), the followng estmaton has been obtaned: deathun 5 = gnpc +.43ltrate (5.93) (.8) The numbers n parentheses, below the estmates, are standard errorss (se) of the estmators. One of the questons posed by researchers s whether ncome has h a negatvee nfluence on nfant mortalty. To answer ths queston, the followng hypothess testng s carred out: The null and alternatvee hypotheses, and the test statstc, are thee followng: H : ˆ.866 t.966 H1 : se( ˆ ).8 Snce the t value s relatvely r hgh, let us start testng wth a level off 1%. For =.1,.1.1 t13 1 t6.39. Gven that t<-.39, as s shown n fgure 4.9, we reect H n favour of H 1. Therefore, the gross natonal ncome per capta has an nfluence that s sgnfcantly s negatve n mortalty of chldren under 5.That s to say, the hgher the gross natonal ncome per capta the lower the percentage of mortalty of chldren under 5.. As H has been reected for =.1, t wll also be reected for levels of 5% and 1%. In the alternatve approach, as can be seen n fgure 4.1, the p-value p correspondng to a t ˆ = for a t wth 61 df s equal to t.. Forr all >., such as.1,.5 and.1, H s reected. (.183) FIGU URE 4.9. Example 4.: Reecton regon usng t wth a left-tal alternatve hypothess. FIGU RE 4.1. Example 4.: p-value usng t wth a left-tal alternatve a hypothess. Two-tal alternatve hypothess Consder now the null hypothess H : 1

11 aganst the alternatve hypothess H : 1 Ths s the relevantt alternatvee when the sgn of s not well determned by theory or common sense. When the alternatve s two-sded, we are nterested n the absolute value of the t statstc. Ths ss a sgnfcance test. In ths case, the decson rule ss the followng: Decson rule If If t ˆ t ˆ t t / nk / nk reect not reect H H (4-14)( Therefore, we reect H : n favor as can be seen n fgure In ths case, n order to reectt H aganstt H : 1, we must obtan a large enough whch s ether postve or negatve. t ˆ of H : 1 at when It s mportant to remark that ass decreases, t ncr reases n absolute value. In the alternatve approach, once the p-value has been determned, we know that whle H s reected for any level of sgnfcance of >p-value, the null hypothess s not reected when <p-value. In I ths case, the p-value s dstrbuted between both tals n a symmetrcal way, as s shown n fguree 4.1. / n k t ˆ /, t nk FIG When a specfc alternatve hypothess s not stated, t s usually consdered to be two-sdethat x s statstcally sgnfcant at thee level hypothess testng. If H s reected n favor of H 1 at a gven, we usually say. EXAM GURE Reecton regon usng t: two-tal alternatve hypothess. MPLE 4.3 Has the rate of crme play a role n the prce of houses n an area? To explan housng prces n an Amercan town, the followng model m s estmated: prce rooms lowstat crmee u 1 wheree rooms s the number of rooms of the house, lowstat s the percentage of people of lower status n the area and crme s crmes commtted per capta n the area. The outpu for the ftted model, usngg the fle hprce (frst 55 observatons), o appears n table 4. and has been taken from E-vews. The meanng of the frst three columns s clear: t-statstc s the outcome to perform a sgnfcance test, that s to say, t s the rato between the Coeffcent and the Std error ; and Prob s the p-value to perform a two-taled test. FIGU 3 URE 4.1. p-value usng t: two-tal alternatve hypothess. 4 11

12 In relaton to ths model, the researcher questonss whether the rate of crmee n an area plays a role n the prce of houses n thatt area. To answer ths queston, the followng procedure has been carred out. In ths case, the null and alternatve hypothess and the test statstc are the followng: H : 4 H : 1 4 ˆ t 4.16 se( ˆ ) 96 4 TABLE 4.. Standardd output n the regresson explanng house prce. n= =55. Varable Coeffcent Std. Error t-statstc Prob. C ROOMS LOWSTAT CRIME Snce the t value s relatvely hgh, let us by start testng wth w a level of 1%. For =.1,.1/.1/ t 51 t (In the usual statstcal tables for t dstrbuton, there s no nformaton for each e df above ). Gven that t.69, we reect H n favour of H 1. Therefore, crme has a sgnfcant nfluence on housng prces for a sgnfcance level of 1% % and, thus, of 5% and 1% %. In the alternatve approach, we can perform the test wth more precson. p In table 4. we see that the p-value for the coeffcent of crme s.. That means that the probablty p of f the t statstcc beng greater than 4.16 s.1 and the probablty of t beng smaller than s.1. That s to say, the p-value, as shown n Fgure 4.13, s dstrbuted n the two tals. As can be seen n ths fgure, H s reected for all sgnfcance levels greater thann., such as.1,.5 and.1. FIGURE Example 4.3: p-value usng t wth a two-tal alternatve hypothess. So far we have seen sgnfcant tests of one-tal and two-tals, n whch a parameter takes the value n H. Now we are gong to look at a more generall case where the parameter n H takes t any value: Thus, the approprate t statstcc s H t ˆ : ˆ se( ˆ ) As before, t ˆ measures how from the hypotheszed value of. many estmated standard devatons ˆ s away EXAM MPLE 4.4 Is the elastcty expendture n frut/ncome equal to 1? Iss frut a luxury good? To answer these questons, we are gong to use the followng model for the expendture n frut: f ln( frut) ln(nc) househsze punders u

13 where nc s dsposable ncome of household, househsze s the number of household members and punder5 s the proporton of chldren under fve n the household. As the varables frut and nc appear expressed n natural logarthms, then s the expendture n frut/ncome elastcty. Usng a sample of 4 households (workfle demand), the results of table 4.3 have been obtaned. TABLE 4.3. Standard output n a regresson explanng expendture n frut. n=4. Varable Coeffcent Std. Error t-statstc Prob. C LN(INC) HOUSEHSIZE PUNDER Is the expendture n frut/ncome elastcty equal to 1? To answer ths queston, the followng procedure has been carred out: In ths case, the null and alternatve hypothess and the test statstc are the followng: H : 1 ˆ ˆ t H1: 1 se( ˆ ˆ.51 ) se( ).1/.1/ For =.1, we fnd that t t. As t >1.69, we reect H. For =.5, t.5/.5/ 36 t As t <.3, we do not reect H for =.5, nor for =.1. Therefore, we reect that the expendture on frut/ncome elastcty s equal to 1 for =.1, but we cannot reect t for =.5, nor for =.1. Is frut a luxury good? Accordng to economc theory, a commodty s a luxury good when ts expendture elastcty wth respect to ncome s hgher than 1. Therefore, to answer to the second queston, and takng nto account that the t statstc s the same, the followng procedure has been carred out: t For =.1, we fnd that t t H : 1 H1: t As t>1.31, we reect H n favour of H 1. For =.5,. As t>1.69, we reect H n favour of H 1. For =.1, t t As t<.44, we do not reect H. Therefore, frut s a luxury good for =.1 and =.5, but we cannot reect H n favour of H 1 for =.1. EXAMPLE 4.5 Is the Madrd stock exchange market effcent? Before answerng ths queston, we wll examne some prevous concepts. The rate of return of an asset over a perod of tme s defned as the percentage change n the value nvested n the asset durng that perod of tme. Let us now consder a specfc asset: a share of an ndustral company acqured n a Spansh stock market at the end of one year and remans untl the end of next year. Those two moments of tme wll be denoted by t-1 and t respectvely. The rate of return of ths acton wthn that year can be expressed by the followng relatonshp: D Pt + Dt + At RA (4-15) t where Pt: s the share prce at the end of perod t, Dt: are the dvdends receved by the share durng the perod t, and At: s the value of the rghts that eventually corresponded to the share durng the perod t Thus, the numerator of (4-15) summarzes the three types of captal gans that have been receved for the mantenance of a share n year t; that s to say, an ncrease or decrease n quotaton, dvdends and rghts on captal ncrease. Dvdng by Pt-1, we obtan the rate of proft on share value at the end of the prevous perod. Of these three components, the most mportant one s the ncrease n quotaton. Consderng only that component, the yeld rate of the acton can be expressed by DP RA (4-16) Pt - 1 t 1t Pt

14 or, alternatvely f we use a relatve rate of varaton, by RA DlnP t (4-17) In the same way as Rat represents the rate of return of a partcular share n ether of the two expressons, we can also calculate the rate of return of all shares lsted n the stock exchange. The latter rate of return, whch wll be denoted by RMt, s called the market rate of return. So far we have consdered the rate of return n a year, but we can also apply expressons such as (4-16), or (4-17), to obtan daly rates of return. It s nterestng to know whether the rates of return n the past are useful for predctng rates of return n the future. Ths queston s related to the concept of market effcency. A market s effcent f prces ncorporate all avalable nformaton, so there s no possblty of makng abnormal profts by usng ths nformaton. In order to test the effcency of a market, we defne the followng model, usng daly rates of return defned by (4-16): rmad9 t 1rmad9t 1 ut (4-18) If a market s effcent, then the parameter of the prevous model must be. Let us now compare whether the Madrd Stock Exchange s effcent as a whole. The model (4-18) has been estmated wth daly data from the Madrd Stock Exchange for 199, usng fle bolmadef. The results obtaned are the followng: rmad 9t rmad9t- 1 (.7) (.69) R =.163 n=47 The results are paradoxcal. On the one hand, the coeffcent of determnaton s very low (.163), whch means that only 1.63% of the total varance of the rate of return s explaned by the prevous day s rate of return. On the other hand, the coeffcent correspondng to the rate of sgnfcance of the prevous day s statstcally sgnfcant at a level of 5% but not at a level of 1% gven that the t.1.1 statstc s equal to.167/.69=., whch s slghtly larger n absolute value than t45 t6 =.. The reason for ths apparent paradox s that the sample sze s very hgh. Thus, although the mpact of the explanatory varable on the endogenous varable s relatvely small (as ndcated by the coeffcent of determnaton), ths fndng s sgnfcant (as evdenced by the statstcal t) because the sample s suffcently large. To answer the queston as to whether the Madrd Stock Exchange s an effcent market, we can say that t s not entrely effcent. However, ths response should be qualfed. In fnancal economcs there s a dependency relatonshp of the rate of return of one day wth respect to the rate correspondng to the prevous day. Ths relatonshp s not very strong, although t s statstcally sgnfcant n many world stock markets due to market frctons. In any case, market players cannot explot ths phenomenon, and thus the market s not neffcent, accordng to the above defnton of the concept of effcency. EXAMPLE 4.6 Is the rate of return of the Madrd Stock Exchange affected by the rate of return of the Tokyo Stock Exchange? The study of the relatonshp between dfferent stock markets (NYSE, Tokyo Stock Exchange Madrd Stock Exchange, London Stock Exchange, etc.) has receved much attenton n recent years due to the greater freedom n the movement of captal and the use of foregn markets to reduce the rsk n portfolo management. Ths s because the absence of perfect market ntegraton allows dversfcaton of rsk. In any case, there s a world trend toward a greater global ntegraton of fnancal markets n general and stock markets n partcular. If markets are effcent, and we have seen n example 4.5 that they are, the nnovatons (new nformaton) wll be reflected n the dfferent markets for a perod of 4 hours. It s mportant to dstngush between two types of nnovatons: a) global nnovatons, whch s news generated around the world and has an nfluence on stock prces n all markets, b) specfc nnovatons, whch s the nformaton generated durng a 4 hour perod and only affects the prce of a partcular market. Thus, nformaton on the evoluton of ol prces can be consdered as a global nnovaton, whle a new fnancal sector regulaton n a country would be consdered a specfc nnovaton. Accordng to the above dscusson, stock prces quoted at a sesson of a partcular stock market are affected by the global nnovatons of a dfferent market whch had closed earler. Thus, global nnovatons ncluded n the Tokyo market wll nfluence the market prces of Madrd on the same day. t 14

15 The followng model shows the transmsson of effects between the Tokyo Stock Exchange and the Madrd Stock Exchange n 199: rmad9 t = 1 + rtok9 t +u t (4-19) where rmad9t s the rate of return of the Madrd Stock Exchange n perod t and rtok9 t s the rate of return of the Tokyo Stock Exchange n perod t. The rates of return have been calculated accordng to (4-16). In the workng fle madtok you can fnd general ndces of the Madrd Stock Exchange and the Tokyo Stock Exchange durng the days both exchanges were open smultaneously n 199. That s, we elmnated observatons for those days when any one of the two stock exchanges was closed. In total, the number of observatons s 34, compared to the 47 and 46 days that the Madrd and Tokyo Stock Exchanges were open. The estmaton of the model (4-19) s as follows: rmad rtok9 t (.7) (.375) R =.45 n=35 Note that the coeffcent of determnaton s relatvely low. However, for testng H : =, the statstc t = (.144/.375) = 3.3, whch mples that we reect the hypothess that the rate of return of the Tokyo Stock Exchange has no effect on the rate of return of the Madrd Stock Exchange, for a sgnfcance level of.1. Once agan we fnd the same apparent paradox whch appeared when we analyzed the effcency of the Madrd Stock Exchange n example 4.5 except for one dfference. In the latter case, the rate of return from the prevous day appeared as sgnfcant due to problems arsng n the elaboraton of the general ndex of the Madrd Stock Exchange. Consequently, the fact that the null hypothess s reected mples that there s emprcal evdence supportng the theory that global nnovatons from the Tokyo Stock Exchange are transmtted to the quotes of the Madrd Stock Exchange that day. 4.. Confdence ntervals Under the CLM, we can easly construct a confdence nterval (CI) for the populaton parameter,. CI are also called nterval estmates because they provde a range of lkely values for, and not ust a pont estmate. The CI s bult n such a way that the unknown parameter s contaned wthn the range of the CI wth a prevously specfed probablty. By usng the fact that bˆ -b tn-k se( bˆ ) ˆ / / Pr tnk tnk 1 se( ˆ ) Operatng to put the unknown alone n the mddle of the nterval, we have ˆ ˆ / ˆ ˆ / Pr se( ) t n k se( ) t nk 1 Therefore, the lower and upper bounds of a (1-) CI respectvely are gven by ˆ ˆ / se( ) t n k ˆ ( ˆ ) / se t n k t 15

16 If random samples were obtaned over and over agan wth, and computed each tme, then the (unknown) populaton value would le n the nterval (, ) for (1 )% of the samples. Unfortunately, for the sngle sample that we use to construct CI, we do not know whether s actually contaned n the nterval. Once a CI s constructed, t s easy to carry out two-taled hypothess tests. If the null hypothess s H : a, then H s reected aganst H : 1 a at (say) the 5% sgnfcance level f, and only f, a s not n the 95% CI. To llustrate ths matter, n fgure 4.14 we constructed confdence ntervals of 9%, 95% and 99%, for the margnal propensty to consumpton - - correspondng to example 4.1.,99,95, FIGURE Confdence ntervals for margnal propensty to consume n example Testng hypotheses about a sngle lnear combnaton of the parameters In many applcatons we are nterested n testng a hypothess nvolvng more than one of the populaton parameters. We can also use the t statstc to test a sngle lnear combnaton of the parameters, where two or more parameters are nvolved. There are two dfferent procedures to perform the test wth a sngle lnear combnaton of parameters. In the frst, the standard error of the lnear combnaton of parameters correspondng to the null hypothess s calculated usng nformaton on the covarance matrx of the estmators. In the second, the model s reparameterzed by ntroducng a new parameter derved from the null hypothess and the reparameterzed model s then estmated; testng for the new parameter ndcates whether the null hypothess s reected or not. The followng example llustrates both procedures. EXAMPLE 4.7 Are there constant returns to scale n the chemcal ndustry? To examne whether there are constant returns to scale n the chemcal sector, we are gong to use the Cobb-Douglas producton functon, gven by ln( output) 1 ln( labor) 3ln( captal) u (4-) In the above model parameters and 3 are elastctes (output/labor and output/captal). Before makng nferences, remember that returns to scale refers to a techncal property of the producton functon examnng changes n output subsequent to a change of the same proporton n all nputs, whch are labor and captal n ths case. If output ncreases by that same proportonal change then there are constant returns to scale. Constant returns to scale mply that f the factors labor and captal ncrease at a certan rate (say 1%), output wll ncrease at the same rate (e.g., 1%). If output ncreases by more than that proporton, there are ncreasng returns to scale. If output ncreases by less than that proportonal change, there are decreasng returns to scale. In the above model, the followng occurs - f + 3 =1, there are constant returns to scale. - f + 3 >1, there are ncreasng returns to scale. - f + 3 <1, there are decreasng returns to scale. 16

17 Data used for ths example are a sample of 7 companes of the prmary metal sector (workfle prodmet), where output s gross value added, labor s a measure of labor nput, and captal s the gross value of plant and equpment. Further detals on constructon of the data are gven n Agner, et al. (1977) and n Hldebrand and Lu (1957); these data were used by Greene n The results obtaned n the estmaton of model (4-), usng any econometrc software avalable, appear n table 4.4. TABLE 4.4. Standard output of the estmaton of the producton functon: model (4-). Varable Coeffcent Std. Error t-statstc Prob. constant ln(labor) ln(captal) To answer the queston posed n ths example, we must test H : 3 1 aganst the followng alternatve hypothess H1: 3 1 Accordng to H, t s stated that 3 1. Therefore, the t statstc must now be based on whether the estmated sum ˆ ˆ 1 3 s suffcently dfferent from to reect H n favor of H 1. Two procedures wll be used to test ths hypothess. In the frst, the covarance matrx of the estmators s used. In the second, the model s reparameterzed by ntroducng a new parameter. Procedure: usng covarance matrx of estmators Accordng to H, t s stated that 3 1. Therefore, the t statstc must now be based on whether the estmated sum ˆ ˆ 1 3 s suffcently dfferent from to reect H n favor of H 1. To account for the samplng error n our estmators, we standardze ths sum by dvdng by ts standard error: ˆ ˆ 3 1 t ˆ ˆ 3 se( ˆ ˆ 3) Therefore, f t ˆ ˆ s large enough, we wll conclude, n a two sde alternatve test, that there are 3 not constant returns to scale. On the other hand, f t ˆ ˆ s postve and large enough, we wll reect, n a 3 one sde alternatve test (rght), H n favour of H1: 3 1. Therefore, there are ncreasng returns to scale. On the other hand, we have se( ˆ ˆ ˆ ˆ 3) var( 3) where var( ˆ ˆ ˆ ˆ ˆ ˆ 3) var( ) var( 3) covar(, 3) Hence, to compute se( ˆ ˆ 3) you need nformaton on the estmated covarance of estmators. Many econometrc software packages (such as e-vews) have an opton to dsplay estmates of the covarance matrx of the estmator vector. In ths case, the covarance matrx obtaned appears n table 4.5. Usng ths nformaton, we have se( ˆ ˆ 3) ˆ ˆ t ˆ ˆ.34 3 se( ˆ ˆ )

18 TABLE 4.5. Covarance matrx n the producton functon. constant ln(labor) ln(captal) constant ln(labor)) ln(captal) Gven that t=.34, t s clear that we cannot reect the exstence of constant returns to scale for the usual sgnfcance levels. Gven that the t statstc s negatve, t makes no sense to test whether there are ncreasng returns to scale Procedure: reparameterzng the model by ntroducng a new parameter It s easer to perform the test f we apply the second procedure. A dfferent model s estmated n ths procedure, whch drectly provdes the standard error of nterest. Thus, let us defne: 3 1 thus, the null hypothess that there are constant returns to scale s equvalent to sayng that H :. From the defnton of we have 3 1. Substtutng n the orgnal equaton: ln( output) 1( 3 1)ln( labor) 3ln( captal) u Hence, ln( output / labor) 1 ln( labor) 3ln( captal / labor) u Therefore, to test whether there are constant returns to scale s equvalent to carryng out a sgnfcance test on the coeffcent of ln(labor) n the prevous model. The strategy of rewrtng the model so that t contans the parameter of nterest works n all cases and s usually easy to mplement. If we apply ths transformaton to ths example, we obtan the results of Table 4.6. As can be seen we obtan the same result: ˆ t ˆ.34 se( ˆ ) TABLE 4.6. Estmaton output for the producton functon: reparameterzed model. Varable Coeffcent Std. Error t-statstc Prob. constant ln(labor) ln(captal/labor) EXAMPLE 4.8 Advertsng or ncentves? The Bush Company s engaged n the sale and dstrbuton of gfts mported from the Near East. The most popular tem n the catalog s the Guantanamo bracelet, whch has some relaxng propertes. The sales agents receve a commsson of 3% of total sales amount. In order to ncrease sales wthout expandng the sales network, the company establshed specal ncentves for those agents who exceeded a sales target durng the last year. Advertsng spots were rado broadcasted n dfferent areas to strengthen the promoton of sales. In those spots specal emphass was placed on hghlghtng the well-beng of wearng a Guantanamo bracelet. The manager of the Bush Company wonders whether a dollar spent on specal ncentves has a hgher ncdence on sales than a dollar spent on advertsng. To answer that queston, the company's econometrcan suggests the followng model to explan sales: sales advert ncent u 1 3 where ncent are ncentves to the salesmen and advert are expendtures n advertsng. The varables sales, ncent and advert are expressed n thousands of dollars. Usng a sample of 18 sale areas (workfle advncen), we have obtaned the output and the covarance matrx of the coeffcents that appear n table 4.7 and n table 4.8 respectvely. 18

19 TABLE 4.7. Standard output of the regresson for example 4.8. Varable Coeffcent Std. Error t-statstc Prob. constant advert ncent TABLE 4.8. Covarance matrx for example 4.8. C ADVERT INCENT constant advert ncent In ths model, the coeffcent ndcates the ncrease n sales produced by a dollar ncrease n spendng on advertsng, whle 3 ndcates the ncrease caused by a dollar ncrease n the specal ncentves, holdng fxed n both cases the other regressor. To answer the queston posed n ths example, the null and the alternatve hypothess are the followng: H : 3 H1: 3 The t statstc s bult usng nformaton about the covarance matrx of the estmators: ˆ 3 ˆ t ˆ ˆ 3 se( ˆ ˆ ) 3 se( ˆ ˆ 3 ) ˆ 3 ˆ t ˆ ˆ se( ˆ ˆ ) For =.1, we fnd that t As t<1.341, we do not reect H for =.1, nor for =.5 or =.1. Therefore, there s no emprcal evdence that a dollar spent on specal ncentves has a hgher ncdence on sales than a dollar spent on advertsng. EXAMPLE 4.9 Testng the hypothess of homogenety n the demand for fsh In the case study n chapter, models for demand for dary products have been estmated from cross-sectonal data, usng dsposable ncome as an explanatory varable. However, the prce of the product tself and, to a greater or lesser extent, the prces of other goods are determnants of the demand. The demand analyss based on cross sectonal data has precsely the lmtaton that t s not possble to examne the effect of prces on demand because prces reman constant, snce the data refer to the same pont n tme. To analyze the effect of prces t s necessary to use tme seres data or, alternatvely, panel data. We wll brefly examne some aspects of the theory of demand for a good and then move to the estmaton of a demand functon wth tme seres data. As a postscrpt to ths case, we wll test one of the hypotheses whch, under certan crcumstances, a theoretcal model must satsfy. The demand for a commodty - say good - can be expressed, accordng to an optmzaton process carred out by the consumer, n terms of dsposable ncome, the prce of the good and the prces of the other goods. Analytcally: q f ( p1, p,, p,, pm, d) (4-1) where - d s the dsposable ncome of the consumer. - p1, p,, p, pm are the prces of the goods whch are taken nto account by consumers when they acqure the good. Logarthmc models are attractve n studes on demand,, because the coeffcents are drectly elastctes. The log model s gven by ln( q ln( p ) ln( p ) ln( p ) ln( p ) ln( R) u (4-) m1 m m 19

20 It s clear to see that all coeffcents, excludng the constant term, are elastctes of dfferent types and therefore are ndependent of the unts of measurement for the varables. When there s no money lluson, f all prces and ncome grow at the same rate, the demand for a good s not affected by these changes. Thus, assumng that prces and ncome are multpled by f the consumer has no money lluson, the followng should be satsfed f ( lp1, lp,, lp,, lpm, lr) f ( p1, p,, p, pm, d) (4-3) From a mathematcal pont of vew, the above condton mples that the demand functon must be homogeneous of degree. Ths condton s called the restrcton of homogenety. Applyng Euler's theorem, the restrcton of homogenety n turn mples that the sum of the demand/ncome elastcty and of all demand/prce elastctes s zero,.e.: m q p h q R (4-4) h1 Ths restrcton appled to the logarthmc model (4-) mples that 3 m 1 m (4-5) In practce, when estmatng a demand functon, the prces of many goods are not ncluded, but only those that are closely related, ether because they are complementary or substtute goods. It s also well known that the budgetary allocaton of spendng s carred out n several stages. Next, the demand for fsh n Span wll be studed by usng a model smlar to (4-). Let us consder that n a frst assgnment, the consumer dstrbutes ts ncome between total consumpton and savngs. In a second stage, the consumpton expendture by functon s performed takng nto account the total consumpton and the relevant prces n each functon. Specfcally, we assume that the only relevant prce n the demand for fsh s the prce of the good (fsh) and the prce of the most mportant substtute (meat). Gven the above consderatons, the followng model s formulated: ln( fsh ln( fshpr) ln( meatpr) ln( cons) u (4-6) where fsh s fsh expendture at constant prces, fshpr s the prce of fsh, meatpr s the prce of meat and cons s total consumpton at constant prces. The workfle fshdem contans nformaton about ths seres for the perod Prces are ndex numbers wth 1986 as a base, and fsh and cons are magntudes at constant prces wth 1986 as a base also. The results of estmatng model (4-6) are as follows: ln( fsh ln( fshpr) ln( meatpr) +.3 ln( cons) (.3) (.133) (.11) (.137) As can be seen, the sgns of the elastctes are correct: the elastcty of demand s negatve wth respect to the prce of the good, whle the elastctes wth respect to the prce of the substtute good and total consumpton are postve In model (4-6) the homogenety restrcton mples the followng null hypothess: 3 4 (4-7) To carry out ths test, we wll use a smlar procedure to the one used n example 4.6. Now, the parameter s defned as follows 3 4 (4-8) Settng 3 4, the followng model has been estmated: ln( fsh 1ln( fshpr) 3ln( meatpr fshpr) 4ln( cons fshpr) u (4-9) The results obtaned were the followng: ln( fsh ln( fshpr ) ln( meatpr ) +.3 ln( cons ) (.3) (.1334) (.11) (.137) Usng (4-8), testng the null hypothess (4-7) s equvalent to testng that the coeffcent of ln(fshpr) n (4-9) s equal to. Snce the t statstc for ths coeffcent s equal to and t.1/ 4 =.8, we reect the hypothess of homogenety regardng the demand for fsh.

21 4..4 Economc mportance versus statstcal sgnfcance Up untl now we have emphaszed statstcal sgnfcance. However, t s mportant to remember that we should pay attenton to the magntude and the sgn of the estmated coeffcent n addton to t statstcs. Statstcal sgnfcance of a varable x s determned entrely by the sze of t ˆ, whereas the economc sgnfcance of a varable s related to the sze (and sgn) of ˆ. Too much focus on statstcal sgnfcance can lead to the false concluson that a varable s mportant for explanng y, even though ts estmated effect s modest. Therefore, even f a varable s statstcally sgnfcant, you need to dscuss the magntude of the estmated coeffcent to get an dea of ts practcal or economc mportance. 4.3 Testng multple lnear restrctons usng the F test. So far, we have only consdered hypotheses nvolvng a sngle restrcton. But frequently, we wsh to test multple hypotheses about the underlyng parameters 1,, 3,, k. In multple lnear restrctons, we wll dstngush three types: excluson restrctons, model sgnfcance and other lnear restrctons Excluson restrctons Null and alternatve hypotheses; unrestrcted and restrcted model We begn wth the leadng case of testng whether a set of ndependent varables has no partal effect on the dependent varable, y. These are called excluson restrctons. Thus, consderng the model y x x x x u (4-3) the null hypothess n a typcal example of excluson restrctons could be the followng: H : 4 5 Ths s an example of a set of multple restrctons, because we are puttng more than one restrcton on the parameters n the above equaton. A test of multple restrctons s called a ont hypothess test. The alternatve hypothess can be expressed n the followng way H 1 : H s not true It s mportant to remark that we test the above H ontly, not ndvdually. Now, we are gong to dstngush between unrestrcted (UR) and restrcted (R) models. The unrestrcted model s the reference model or ntal model. In ths example the unrestrcted model s the model gven n (4-3). The restrcted model s obtaned by mposng H on the orgnal model. In the above example, the restrcted model s y x x u By defnton, the restrcted model always has fewer parameters than the unrestrcted one. Moreover, t s always true that 1

22 RSS R RSS UR where RSS R s the RSS of the restrcted model, and RSS UR s the RSS of the unrestrcted model. Remember that, because OLS estmates are chosen to mnmze the sum of squared resduals, the RSS never decreases (and generally ncreases) when certan restrctons (such as droppng varables) are ntroduced nto the model. The ncrease n the RSS when the restrctons are mposed can tell us somethng about the lkely truth of H. If we obtan a large ncrease, ths s evdence aganst H, and ths hypothess wll be reected. If the ncrease s small, ths s not evdence aganst H, and ths hypothess wll not be reected. The queston s therefore whether the observed ncrease n the RSS when the restrctons are mposed s large enough, relatve to the RSS n the unrestrcted model, to warrant reectng H. The answer depends on but we cannot carry out the test at a chosen untl we have a statstc whose dstrbuton s known, and s tabulated, under H. Thus, we need a way to combne the nformaton n RSS R and RSS UR to obtan a test statstc wth a known dstrbuton under H. Now, let us look at the general case, where the unrestrcted model s y x x x u (4-31) k k+ Let us suppose that there are q excluson restrctons to test. H states that q of the varables have zero coeffcents. Assumng that they are the last q varables, H s stated as H : (4-3) kq1 kq k The restrcted model s obtaned by mposng the q restrctons of H on the unrestrcted model. y x x k qxk q u (4-33) H 1 s stated as H 1 : H s not true (4-34) Test statstc: F rato The F statstc, or F rato, s defned by ( RSSR RSSUR)/ q F RSS /( n k) UR (4-35) where RSS R s the RSS of the restrcted model, and RSS UR s the RSS of the unrestrcted model and q s the number of restrctons; that s to say, the number of equaltes n the null hypothess. In order to use the F statstc for a hypothess testng, we have to know ts samplng dstrbuton under H n order to choose the value c for a gven, and determne the reecton rule. It can be shown that, under H, and assumng the CLM assumptons hold, the F statstc s dstrbuted as a Snedecor s F random varable wth (q,n-k) df. We wrte ths result as F H F - (4-36) qn, k