Lecture Notes Types of economic variables

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1 Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte umber of elemets or a ftely coutable umber, such as all postve tegers. Example: Number of workers, etc. () Categorcal data are grouped accordgly to some qualty or attrbute Example: Sex or type of automoble.. Revew of statstcs () Populato the total group set of elemets of terest. Sample a subset of the populato. We usually collect samples because t s too costly to sample the etre populato. Example College studets survey Kazakhsta Populato s all college studets Probablty the relatve frequecy or occurrece of a evet after repettve trals or expermets. Probablty les betwee 0 ad 1 All probabltes for all evets have to sum to 1 Example: 60% chace of ra today Imples a 40% of o ra, summg to oe or 100% 1

2 () Probablty dstrbuto fuctos (PDF) a fucto that assocates each value of a dscrete radom varable wth the probablty that ths value wll occur. Deoted as p(x) or f(x) Cumulatve probablty dstrbuto fucto (CDF) - tegral of a probablty fucto Deoted by a captal letter, such as P(x) or F(x). P x x f t dt If you sum over all probabltes, the t has to equal oe. P x f t dt 1 The ormal probablty dstrbuto s show below

3 () Descrptve statstcal measures descrbe a sample or populato. Measures of cetral tedeces Mea calculated by x x f (x ) where f(x) s the pdf ad x s the radom varable. Ths s also the expected value If every observato s equally lkely, the s the umber of observatos 3 1 x x where Meda the mddle pot or observato whe the data are ordered from smallest to largest. Mode the value, whch occurs most ofte a dstrbuto. The peak of a dstrbuto Use calculus to fd maxmum value Rage the dfferece betwee the largest value the sample (the maxmum) ad the smallest value (the mmum), R x max x m Varace s a measure of devato from the mea, deoted by Varace has a problem If the uts are $ s, the varace s $ The -1 s the sample varace Degrees of freedom (df) the amout of formato you have,.e. the umber of observatos Sce you estmated the varace, you lose oe pece of formato I regresso, you have k parameters df = k, because you estmated k stadard errors

4 1 ˆ ( x x) 1 where x deotes the mea of the sample Stadard devato the varablty or spread of the data aroud ts mea Has same uts of the varable (.e. mles, dollars, klometers, etc.). ˆ ˆ (v) The Normal Dstrbuto or Gaussa Dstrbuto the most commo dstrbuto Bell-shaped curve Regresso you do ot eed a dstrbuto to estmate parameters However, f are testg the statstcal sgfcace of a parameter, the you eed a dstrbuto. The top dstrbuto s symmetrc, thus the mea = meda = mode 4

5 The bottom dstrbutos are ot symmetrc, so mea meda mod e The Normal Dstrbuto We have a relatoshp betwee probabltes ad the mea ad stadard devato Wrte t as ~ Nx, x Statstcas have a trck to make all ormal dstrbutos stadard, whch s ~ N0,1 z Dstrbuto s below: 5

6 The trasformato s: z x x I the old days, people carred tables that had the probablty for partcular z values. Excel ca calculate ths easly =ormdst(z) Note ths returs the probablty from egatve fty to the z value (v) Stadard Error the varablty the mea whe takg repeated samples The true parameter s ukow, Each tme I take a dfferet sample from the populato, I get a dfferet estmate for the parameter Example Mea of frst sample s 3.5 Mea of secod sample s 4. Mea of thrd sample s.9 std. error Sce I caot keep takg samples, or ca I take observatos of the whole populato, I would lke to kow how my estmator vares. ˆ z ˆ Very smlar to the z-trasformato. I ca use ths relatoshp for hypothess testg. Each hypotheses has two parts Null - s the hypothess of terest, H 0 Alteratve - s the complemet of the ull, H A 6

7 A hypothess has to corporate all possble outcomes Example H 0 : = 4 H A : > 4 What about values below 4? Properly stated ull ad alteratve hypothess cover all alteratves. Example Two-tal test H 0 : = 0 H A : 0 Ths s very commo Automatcally computed for Lear Regresso Does t appear the parameter equals zero? The x varable has o mpact o the y Example Two-tal test H 0 : < 0 H A : 0 We have a problem, we do ot kow what the stadard devato s ad have to use a dfferet dstrbuto (v) t-dstrbuto a symmetrc bell-shaped dstrbuto The problem s we do ot kow the parameter, Further, we do ot kow the stadard devato too,.e. A lttle fatter ad shorter tha the stadard bell curve The shape depeds o degrees of freedom 7

8 Degrees of freedom s the amout of formato,.e. umber of observatos We had to estmate the stadard devato, so we subtract 1 Regresso each parameter estmate has a stadard error k parameters, thus, df = k As the degrees of freedom approach fty, the t- dstrbuto approaches the stadard ormal dstrbuto. ˆ The t-test s ˆ t ˆ 8 ˆ where hat s the varable of terest s the ull hypothess value ˆ s a approprate estmate of the stadard devato of x s the umber of observatos. Example Homework #1 Does the data support that =? We choose a level of sgfcace, Usually = 0.05 Two-tal test H 0 : = - H A : - A two-tal test. Now calculate the degrees of freedom 60 observatos ad estmated two parameters df= 58 Fd crtcal value for t value.

9 Remember t s a two-tal test, so put half alpha to each tal Be careful wth Excel: Use Excel =tv(0.05, 58) It returs t c =.00 The c s for crtcal value Now calculate the t-statstc From regresso output, calculate the stadard error Std. error = 0.0 Parameter estmate for b = -0.5 ˆ t ˆ Reject the H 0 f t >.00 or t < -.00 Fal to reject f -.00 < t <.00 Reject the H 0 ad coclude that the parameter estmate does ot equals -. Let s do the most commo Two-tal test H 0 : = 0 H A : 0 ˆ t ˆ Reject the H 0 ad coclude that the parameter estmate does ot equal zero 9

10 Selected Crtcal Values for the t-dstrbuto Level of Sgfcace α - see dagrams above Degrees of Freedom Aalyss of Varace (ANOVA) I terms of regressos, ANOVA s used to test hypothess may types of statstcal aalyss Sum of Squared Total (SST) s defed as: 10 SST y s the depedet varable the regresso The y y s the total varato for observato Sum of Squared Regresso (SSR) s defed as: Ths s the varato explaed by the regresso 1 ( y y). SSR (ŷ y). Sum of Squared Errors (SSE), whch was earler defed as: SSE 1 y yˆ. uˆ. 1 SSE s the amout of varato ot explaed by the regresso equato. Thus, SST = SSR + SSE, whch s proved the chapter 1

11 We ca use ths formato to calculate the R statstc: Show relatoshp: SST SSR SSE SSR SSE 1 SST SST SSR SSE R 1 SST SST R 1 SSE SST Problem the more parameters added to the regresso, the hgher the R. R = 1, f = k, the umber of parameters equal observatos Now we eed the degrees of freedom for each measure: Sum of Squared Regresso (SSR) df =k 1 Sum of Squared Errors (SSE) df = k Sum of Squared Total (SST) df = 1 We calculate the Mea Square (MS) Regresso (MS) = SSR / (k 1) Resdual (MS) =SSE / ( k) Total (MS) NA Addtoal formato Whe you have a varable wth a ormal dstrbuto If you add or subtract f from other varables wth a ormal dstrbuto, the t s stll ormally dstrbuted Calculatg a mea s a frst momet If you square a radom varable wth a ormal dstrbuto, the you get a ch-square dstrbuto wth degrees of freedom. The squares are varaces ad called the secod momet All the Mea Squares are dstrbuted as ch squares 11

12 F-dstrbuto ca test a whole group of hypothess or test a whole regresso model F- test ca test may other thgs The F-test s a rato of two ch-squares The F-test s a oe-taled test assocated wth the rght-had tal. Squarg makes all terms postve The F-dstrbuto ad test s as follows: H 0 : Regresso model does ot expla the data,.e. all the parameters estmates are zero H A : Regresso model does expla the model,.e. at least oe parameter estmate s ot zero Frst, we eed the crtcal value: a = 0.05, df 1 = 1, ad df =58 I Excel, =fv(0.05,1,58) F c = 4.00 Excel calculates the ANOVA ANOVA df SS MS F Sgfcace F Regresso Resdual Total

13 SSR df SSE df Calculate the F-value = The computed F exceeds the F c, so reject the H 0, ad coclude at least oe parameter s ot equal to zero. Example from homework #3. How may observatos? 10 How may parameters, k? 4 Degrees of freedom for error df = 10 4 = 6 Degrees of freedom for total df = 10 1 = 9 ANOVA df SS MS F Sgfcace F Regresso E-06 Resdual Total

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot