Onlne Appendx to: Axomatzaton and measurement of Quas-hyperbolc Dscountng José Lus Montel Olea Tomasz Strzaleck 1 Sample Selecton As dscussed before our ntal sample conssts of two groups of subjects. Group M has 639 subjects that answered the Money questonnare. Group IC has 64 subjects that answered the Ice-cream questonnare. Insde each group, we assocate subjects wth an Internet Protocol address (IP) and we verfy that there s no IP repetton nsde the group. Consequently, we do not allow for a sngle IP address to answer the same questonnare more than once. Remark 1. We do allow for the same IP address to answer both questonnares. Our sample contans 548 IPs n ths stuaton. Note that ths wthn-subject nformaton could be useful for partally dentfyng the jont and condtonal dstrbutons of money/ce-cream preference parameters. For nstance, let β X We could try to understand whether or not β M denote the short-run dscount factor for a good X related queston. and β IC are ndependent; ths s: { } } { } µ P β M β 1, β IC β 2 = ( ) µ { P β M β 1 µ P β IC β 2 We left these (and other related) questons for future research and we focus on nference concernng the dstrbuton of (β X, δ X ) for a fxed good X.
1.1 Monotoncty and Understandng We checked the subjects understandng of the nstructons by askng two smple questons, see Fgure 1. Table 1 summarzes our sample selecton based on the two questons reported n Fgure 1: Fgure 1: Intal Checks Unque IPs Survve u Check Survve m Check Survve m-u Check Money 639 68 526 56 (79.1%) IC 64 611 526 57 (79.2%) NOTE: m stands for monotoncty and u stands for understandng Table 1: Sample Selecton based on m-u check 2
1.2 Consstency After selectng subjects that survve the monotoncty and understandng check, we further refne the sample by consderng agents that are consstent wth the quas-hyperbolc model. A necessary condton for an agent to admt a quas hyperbolc representaton s the exstence of at most one swtch pont (from patent to mpatent prospect) per prce lst. Thus, we dscard all subjects that volate ths condton. Sample after m-u check Inconsstent (L1) Inconsstent (L2) Consstent Money 56 36 156 336 IC 57 5 31 444 Table 2: Sample Selecton based on consstency check We also summarze the type of nconsstency observed n each prce lst. The hstograms n Fgure 2 report the frequency of swtchng ponts. Note that agents wth a sngle swtch pont movng from an mpatent reward n queston j to a patent reward n j + h are also nconsstent. A quck observaton concernng the behavor of nconsstent subjects. It seems reasonable to ask whether subjects wth nconsstent answers n prce lst 1 also have nconsstent answers n prce lst 2. One way to get a smple statstc to summarze ths dependence s as follows. Consder frst the money questonnare. For prce lst 1, we create a vector of dummy varables d 1 (of dmenson 56) wth value of 1 f the agent s nconsstent, and zero otherwse. The dummy varable d 2 s defned analogously. We then look at the sample correlaton between these two random vectors. The correlaton equals 1 f and only f the subjects that are nconsstent n the prce lst 1 are also nconsstent n prce lst 2. Lkewse, the correlaton equals f and only f there s no overlap. For the money questonnare, we found a correlaton of.1815; for the ce-cream questonnare the correlaton s.4126. 3
12 Prce Lst 1 12 Prce Lst 1 1 1 8 8 Frequency 6 Frequency 6 4 4 2 2 1 2 3 4 5 6 Swtch Ponts (a) Money 1 2 3 4 5 6 Swtch Ponts (b) Ice-cream 12 Prce Lst 2 12 Prce Lst 2 1 1 8 8 Frequency 6 Frequency 6 4 4 2 2 1 2 3 4 5 6 Swtch Ponts (c) Money 1 2 3 4 5 6 Swtch Ponts (d) Ice-cream Fgure 2: Dstrbuton of Swtch Ponts for Inconsstent Subjects 2 Response Tmes Gven the onlne nature of our plot experment and the lack of ncentves, a concern s that subjects clck at random, or always choose the same answer (for example always choose A) n order to save tme and move quckly to another task. We fnd lttle support of ths story n the data. I ths secton, we descrbe the data collected on response tmes and the statstcal test we mplemented to show that the populaton s upper and lower bounds are statstcally ndependent 4
of response tmes. 2.1 Data on Response Tmes For each prce lst 1-2 n the M-IC questonnares, we collected three varables measurng subjects response tmes: 1. r : The total tme spent n questonnare M (IC), measured as total number of seconds that each subject spent n completng the two prce lsts. 2. r,1 : The tme spent n prce lst 1 of questonnare M (IC), measured as total number of seconds that each subject spent n completng the frst prce lst of of questonnare M (IC). 3. r,2 : The tme spent n prce lst 2 of questonnare M (IC), measured as total number of seconds that each subject spent n completng the second prce lst of of questonnare M (IC)..45.4 Total Tme Dstrbuton per Category Always Left/Rght Others.5.45 Total Tme Dstrbuton per Category Always Left/Rght Others.35.4 (%)Percentage of people n each group.3.25.2.15 (%)Percentage of people n each group.35.3.25.2.15.1.1.5.5 2 4 6 8 1 12 14 16 18 2 Mnutes (a) Money 2 4 6 8 1 12 14 16 18 2 Mnutes (b) Ice-cream Fgure 3: Condtonal Dstrbutons of Total Response Tme Fgure 3 compares the total response tmes r for consstent subjects that always select Opton A (s,1 = s,2 = 8) or Opton B (s,1 = s,2 = 1) aganst consstent subjects wth other behavor. 5
The dstrbutons n Fgure 3 are condtonal dstrbutons of response tmes for certan values of (s,1, s,2 ): and r (s,1 = s 1,2 = 1 or s,1 = s 1,2 = 8) r (s,1 = s 1,2 = 1 or s,1 = s 1,2 = 8) c Fgure 4 below reports the condtonal dstrbutons of response tmes gven s,1 = k (frst row) and gven s,2 = k (second row). Both graphs suggest that the response tme r s ndependent of both s,1 and s,2. We test ths statstcal hypothess usng the dstance covarance statstc of?. The dstance covarance statstc compares the weghted dfference between the sample analog of the characterstc functon of (s,1, s,2, r ) aganst the product of the characterstc functons of (s,1, s,2 ) and r, see?, pp. 6-7. Under the null hypothess of ndependence, the (properly scaled) dstance covarance between (s,1, s,2, r ) and r converges n dstrbuton to a weghted sum of ch-squared random varables and a 5%-level conservatve crtcal value s gven 1.96 2 ; see Theorem 5, 6 n?. The scaled dstance covarance statstc s 1.3 for the M questonnare and.851 for IC. In both cases, the conservatve 5% crtcal value s 1.96 2. Thus, we cannot reject the null hypothess that the dstrbutons of (s,1, s,2 ) and r are ndependent. 1 The dstance correlaton (normalzed to be n [,1]) s.127 for the Money questonnare and.75 for the Ice-cream. The dstance correlaton s zero n the populaton f and only the random vectors are ndependent. The populaton lower and upper bounds n our desgn are functons of the swtch ponts (s,1, s,2 ). If (s,1, s,2 ) and r are ndependent then: µ{ P s,1 k, s,2 = j + 1, r > r}/µ{ P r > r} 1 The dstance covarance statstc was computed usng the matlab fle dstcorr.m avalable here: http://www. mathworks.com/matlabcentral/fleexchange/3995-dstance-correlaton/content/dstcorr.m 6
Tme Dstrbuton (Prce Lst 1) Money Year FR FC 1:Very Impatent, 8:Very Patent Tme Dstrbuton (Prce Lst 1) IC Year FR FC 1:Very Impatent, 8:Very Patent 3 3 25 25 2 2 Mnutes 15 Mnutes 15 1 1 5 5 Categores (a) Money Categores (b) Ice-cream Tme Dstrbuton (Prce Lst 2) Money Year FR FC 1:Very Impatent, 8:Very Patent Tme Dstrbuton (Prce Lst 2) IC Year FR FC 1:Very Impatent, 8:Very Patent 3 3 25 25 2 2 Mnutes 15 Mnutes 15 1 1 5 5 Categores (c) Money Categores (d) Ice-cream Fgure 4: Condtonal Dstrbutons of Response Tme by Swtch Pont Category equals µ{ P s,1 k, s,2 = j + 1} for all c. We conclude by sayng that there s no statstcal evdence suggestng that the lower and upper bounds wll change f we condton on response tmes. 7
3 Worker s Qualfcatons In terms of qualfcatons, we dvde the subjects n our sample nto Masters (MA) and Non- Masters wth qualfcatons (NMAQ). AMT defnes Masters as an elte groups of workers who have demonstrated accuracy on specfc types of HITs on the Mechancal Turk marketplace. Workers acheve a Masters dstncton by consstently completng HITs of a certan type wth a hgh degree of accuracy. For non masters, AMT allows the users to requre dfferent degrees of qualfcatons. A qualfcaton represents a worker s skll, ablty or reputaton. The NonMasters wth qualfcatons subjects n our sample are workers wth 95% of approved pror tasks and at mnmum 5 approved pror tasks. In ths secton, we analyze the number of MA and NMAQ n our sample. We also dscuss the dependence of swtch ponts to ths categorzaton of workers. In partcular, we reject the null hypothess that the dstrbuton of swtch ponts n the populaton s ndependent of the MA/NMAQ category dummy. Fnally, we report lower and upper bounds for MA and NMAQ. 3.1 MA and NMAQ n our sample Sample m-u-check MA m-u sample Consstent MA Consstent NMAQ Money 56 144 88 248 IC 57 142 118 326 Table 3: Sample Selecton based on consstency check Table 3 reports the number of MA and NMAQ workers n our sample. We observe a larger share of NMAQ (the rato s almost 3:1) n both the Money and Ice-cream treatments. Fgure 5 presents the condtonal dstrbuton of swtch ponts n prce lst 1 for the money and ce-cream treatments. Panel a) of ths fgure suggests that the dstrbuton of swtch ponts n queston 1 s not ndependent of the MA/NMAQ dummy varable (d MA ): the condtonal probablty of 8
never swtchng (category 8) s larger for non-masters. Interestngly, Panel b) suggests that the condtonal dstrbuton of swtch ponts n prce lst 1 for the ce-cream treatment does not vary n the MA/NMAQ groups. At the end of ths secton we wll provde statstcal tests for the null hypothess of ndependence for the random varables s,1 and d MA. We wll also report the estmated bounds for G(β) for the MA and NMAQ..35 Masters Non Masters Dstrbuton of Swtch Ponts (Queston 1).4 Masters Non Masters Dstrbuton of Swtch Ponts (Queston 1).3.35 (%)Percentage of people n each group.25.2.15.1 (%)Percentage of people n each group.3.25.2.15.1.5.5 Swtch Pont Category (a) Money Swtch Pont Category (b) Ice-cream Fgure 5: Condtonal Dstrbutons of Swtch Pont Fgure 6 presents the condtonal dstrbuton of swtch ponts n prce lst 2 for the money and ce-cream treatments. The graphs suggest that the condtonal dstrbutons of s,1 d MA s,1 d MA = are smlar. = 1 and We now consder the tests for three dfferent null hypothess (each of them tested n the money and ce-cream treatment separately). 1. H 1 : s,1 s ndependent of d MA 2. H 2 : s,2 s ndependent of d MA vs. H 1 : s,1 s not ndependent of d MA vs. H 1 : s,2 s not ndependent of d MA 3. H 3 : (s,1, s,2 ) s ndependent of d MA vs. H 1 : (s,1, s,2 ) s not ndependent of d MA Once agan, we test these statstcal hypotheses usng the dstance covarance statstc of? dscussed n Appendx 2. The followng table reports the (properly scaled) dstance covarance 9
.35 Masters Non Masters Dstrbuton of Swtch Ponts (Queston 2).35 Masters Non Masters Dstrbuton of Swtch Ponts (Queston 2).3.3 (%)Percentage of people n each group.25.2.15.1 (%)Percentage of people n each group.25.2.15.1.5.5 Swtch Pont Category (a) Money Swtch Pont Category (b) Ice-cream Fgure 6: Condtonal Dstrbutons of Swtch Pont statstc (see pp. 6-7 n?). The null hypothess of ndependence s rejected at the 5% asymptotc level f the scaled dstance covarance statstc s larger than 1.96 2 = 3.84. We also report the dstance correlaton (normalzed to be n [, 1]) as a measure of dependence. H 1 Money H 2 H 3 Dstance Correlaton.197.63.169 Dstance Covarance statstc 9.38.958 5.43 IC Dstance Correlaton.428.55.54 Dstance Covarance statstc.61.992.864 Table 4: Dstance Correlaton and Dstance Covarance statstcs For the money treatment, H 1 s rejected at the 5% asymptotc level as the dstance covarance statstc s larger than 3.84. Therefore, we reject the null hypothess that s,1 s ndependent of d MA. For the same treatment, we cannot reject the null hypothess H 2. The latter suggests that 1
the populatons bounds for F (δ) do not depend on whether we condton on MA/NMAQ. For the IC treatment, we cannot reject H 1 at the 5% asymptotc level as the dstance covarance statstc s no larger than 3.84. Lkewse, the null hypotheses H 2, H 3. 3.2 Lower and upper bounds for MA/NMAQ Snce H 2 s rejected for both the money and the ce-cream treatments, the populaton bounds for F (δ d MA ) should not depend on whether we condton on MA or NMAQ. The same s true for the bounds for G(β d MA ) n the ce-cream treatment. However, for the money treatment we could expect the bounds for G(β d MA the results. ) to depend on the values of d MA. Fgure 7 and 8 present 1 Bounds for the c.d.f. of β (Masters): Set Estmators 1 Bounds for the c.d.f. of β (Non Masters): Set Estmators.9.9.8.8.7.7.6.6 F(β).5 F(β).5.4.4.3.3.2.2.1.1.5 1 1.5 2 2.5 3 β.5 1 1.5 2 2.5 3 β (a) MA (b) NMAQ Fgure 7: Condtonal Lower and Upper Bounds for G(β d MA ): Money 4 Jont Dstrbutons The prevous secton focused on the partal dentfcaton of the margnal dstrbutons of β and δ for two dfferent rewards. We know dscuss the fndngs concernng the jont dstrbutons of (β, δ ) and (β M, β IC ). 11
1 Bounds for the c.d.f. of β (Masters): Set Estmators 1 Bounds for the c.d.f. of β (Non Masters): Set Estmators.9.9.8.8.7.7.6.6 F(β).5 F(β).5.4.4.3.3.2.2.1.1.5 1 1.5 2 2.5 3 β (a) MA.5 1 1.5 2 2.5 3 β (b) NMAQ Fgure 8: Condtonal Lower and Upper Bounds for G(β d MA ): Ice-Cream 4.1 Jont dstrbuton of (β, δ ) Our expermental desgn allows us to partally dentfy the jont dstrbuton of (β, δ ). For nstance, note that: µ{ P β 1 and δ δ (j)} j µ{ P s,1 < k and s,2 = k} k=1 and µ{ P β 1 and δ δ (j)} j µ{ P s,1 k + 1 and s,2 = k} k=1 One queston we could ask concernng the jont dstrbuton of β and δ s whether the tme preference parameters are ndependent n the populaton. Although we are not aware of statstcal tests for the ndependence of two random varables whose jont (and margnals) c.d.f s are partally dentfed, we present a smple analyss that can shed some lght on the ssue. 12
Note that under the assumpton of ndependence, the followng upper and lower bounds obtan: and µ{ P β 1 and δ δ (j)} = G(1)F (δ (j)) ( 8 ) µ{ P s,1 < j and s,2 = j} F (δ (j)) j=1 µ{ P β 1 and δ δ (j)} = G(1)F (δ (j)) ( 8 ) µ{ P s,1 j + 1 and s,2 = j} F (δ (j)) j=1 Fgure 9 presents upper and lower bounds for µ{ P β 1 and δ δ} as a functon of δ [.6, 1]. The fgure suggests that regardless of statstcal sgnfcance, the dfference that arses from the ndependence assumpton does not seem to be very mportant, at least when the jont c.d.f. s evaluated at β = 1. 1 Bounds wthout Independence 1 Bounds wthout Independence.9 Bounds wth Independence.9 Bounds wth Independence.8.8.7.7.6.6.5.5.4.4.3.3.2.2.1.1.65.7.75.8.85.9.95 1 δ.65.7.75.8.85.9.95 1 δ (a) Money (b) Ice-cream Fgure 9: Lower and Upper bounds for µ{ β 1 and δ δ} 13
4.2 Jont dstrbuton of preference parameters for dfferent prmary rewards As mentoned before, there are 548 partcpants that answered both the money and ce-cream questonnare. Out of those, there are 437 that survve the m-u check and 273 that survve the m-u-c check. In prncple, one could use the nformaton concernng the swtch ponts n the 4 prce lsts (2 prce lsts per questonnare) to bound probablty statements concernng preference parameters for dfferent rewards; for example, the probablty of the event: { β M 1 and β IC 1} The probablty of ths event cannot be bounded drectly usng the results concernng the margnals β M and β IC, as these dstrbutons need not be ndependent. Thus, the frst queston that we ask s whether there s dependence between the vectors (s M,1, s M,2) and (s IC,1, s IC,2 ). A smple statstc to report s the correlaton matrx between the 4 random vectors (s M,1, s M,2, s IC,1, s IC,2 ): 1.772.4734.415.772 1.5914.5267.4734.5914 1.8847.415.5267.8847 1 The matrx above suggests that there s dependence between swtch ponts wthn the same questonnare, but also across questonnares. If ths dependence s also present n the swtch ponts assocated to more complcated annutes (such as those requred by our annuty compensaton axom), then we could expect the preference parameters across dfferent prmary rewards to exhbt dependence as well. Thus, we test the null hypothess: H : (s M,1, s M,2) s ndependent of (s IC,1, s IC,2 ) aganst the alternatve that the random vectors are not ndependent. The dstance correlaton of? s.568 and ther test statstc for ndependence s 31.9643. Snce the 5% asymptotcally vald 14
crtcal value s 3.84, the null hypothess of ndependence s rejected. Fnally, we report the share of agents (out of those that solved both questons) for whch s M,1 < s M,2 and s IC,1 < s IC,2. Ths s, we report the share of agents that exhbt behavor consstent wth present bas n both questonnares. The share of agents that exhbt present bas s 12.82% n the money questonnare and 11.36% n the IC questonnare. The share of agents that exhbt present bas n both questonnares s only 1.47%. 15