Context Matters : Empirical Analysis in Social Science, Where the Effect of Anything Depends on Everything Else

1. Context Matters:Empirical Analysis in Social Science,Where the Effect of AnythingDepends on Everything Else Interaction Terms, Multilevel Models, & Nonlinear Regression Blalock Lecture ICPSR Summer School, 30 July 2009 (Here for fuller pedagogical slides & materials: http://www.umich.edu/~franzese/SyllabiEtc.html) Robert J. Franzese, Jr. Professor of Political Science, The University of Michigan, Ann Arbor

2. Overview • Interactions in Pol-Sci: Ubiquitous, but more • The Linear-Interaction Model • From theory to empirical-model specification: Arg’s that imply interax (& some that don’t), & how write. • Interpretation: • Effects=derivatives & differences, not coefficients! • Std Errs (etc.): effects vary, so do std errs (etc.)! • Presentation: Tables & Graphs, & Choosing Specifications • Elaborations, Complications, & Extensions: • Use & abuse of some common-practice “rules” • Multilevel/Hierarchical Issues: • Sample-Splitting v. (Dummy-) Interacting • Variances in Interax, & Rndm-effect/Hierarchical/Multilevel, Models • Interax in QualDep (inherent-interax, latent-var) models • Complex Context-Conditionality & Nonlinear Least-Squares

3. Interactions in Pol-Sci Research • Common. ‘96-‘01 AJPS, APSR, JoP: • 54% some stat meth (=s.e.’s), of which 24% = interax (so interax ≈ 12.5% or 1/8th total). • (N.b., most rest QualDep & formal theory, not counted, so understate tech nature of discipline)

4. Interactions in Pol-Sci Theory1 • Our Theories/Substance say should be more; core classes of argument almost inherently interactive: • INSTITUTIONAL: institutions=interactive variables: • funnel, moderate, shape, condition, constrain, refract, magnify, augment, dampen, mitigate political processes that… • …translate societal interest-structure into effective political pressures, • &/or pressures into public-policy responses, • &/or policies to outcomes. I.e., they affect effectsinteraction. • Views from across institutionalist perspectives: • Hall: “institutionalist model=>policy more than sum countervailing pressure from soc grps; that press mediated by organizational dynamic.” • Ikenberry: “[Political struggles] mediated by inst’l setting where [occur]” • Steinmo & Thelen: “inst’s…constrain & refract politics… [effects of] macro-structures magnify or mitigated by intermediate-level inst’s… help us…explain the contingent nature of pol-econ development…” • Shepsle: “SIE clearly a move [to] incorporating inst’l features into R-C. Structure & procedure combine w/ preferences to produce outcomes.”

5. Interactions in Pol-Sci Theory2 • Thry/Subst: core classes arg inherently interax (cont.): • INSTITUTIONAL: … • STRATEGIC: actors’ choices (outcomes) conditional upon inst’l/struct’l environ., opp.-set, & other actors’ choices. • CONTEXTUAL: actors’ choices (outcomes) conditional upon environ., opp. set, & aggregates of other actors’ choices. • Across all subfields & areas: • Comp Pol… Int’l Rltns… U.S. Pol… For examples: • Electoral system & societal structure  party system. • System polarity & offense-defense balance war propensity. • Divided government & polarization  legislative productivity. • PolEcon… PolBehave… LegStuds… PolDev… • Electoral & partisan cycles depend on inst’l & econ conditions • Gov’t inst’s shape voter behavior: balancing (Kedar, Alesina); economic voting (Powell & Whitten); etc. • Effects divided gov’t depend presidential v. parliamentary. • Effect inequality on democratization depends cleavage structure.

6. Interactions in Political Science Theory & Substance:Everyone’s Favorite “Model”

7. Interactions in Political Science Theory & Substance:An Old (& still) Favorite “Model” of Mine

8. Interactions in Political Science Theory & Substance:An Newer Favorite “Model” of Mine • Complex Context-Conditionality: • Effect of (almost) everything depends on (almost) everything else. • E.g., Principal-Agent Situations • If fully principal, y1=f(X); if fully agent, y2=g(Z); institutions: 0≤h(I)≤1.

9. (Complex) Context-Conditionality: (Hallmark of Modern Pol-Sci Theory?) • Principal-Agent (Shared Control) Situations, for example: • If fully principal: y1=f(X); • If fully agent: y2=g(Z); • Institutions=>Monitoring & Enforcement costs principal must pay to induce agent behave as principal would: 0≤h(I)≤1. • RESULT: • In words… … … …i.e., effect of anything depends on everything else!

10. Not Every Argument Is an Interactive Argument • Not Interactive: • X affects Y through its effect on Z: XZY • In (political) psychology / behavior, this called mediation. Interaction is called moderation in this literature. • X and Z affect each other: XZ. • I.e., X and Zendogenous to each other. Note: irrelevant to Gauss-Markov (OLS is BLUE); merely implies care to what partials (coefficients) mean. • Y depends on X controlling for Z, or Y depends on X & Z: E(Y|X,Z)=f(Z), E(Y|X)=f(Z), Y=f(X,Z) • I.e., the outcomes differ across 22 of X and Z. • Interactive: Effect of X on Y depends on Z ( converse: Effect of Z on Y depends on X):

11. From Theory/Substance to Empirical-Model Specification • Classic Comparative-Politics Example: • Societal Fragmentation, SFrag, & • Electoral-System Proportionality, DMag, • Effective # Parliamentary Parties: ENPP • “Theory”: • Hypotheses: • Empirical Specification: Lots ways get there...

12. A Typical Linear-Interactive Specification • Want linear f() w/ these properties; many ways to get there:

13. Interpretation of Effects:Derivatives & Differences, Not Coefficients • Standard Linear Interactive Model: • Effect of SFrag on ENPP (is a function of DMag): • Effect of DMag on ENPP (is f of SFrag):

14. Interpretation of Effects: NOTES1 • “Main Effect”: e.g., SF = “main effect of SFrag” • ….butSF is merely the effect of SFrag at other variable(s) involved in interaction with it=0, so: • Other-var(s)=0 may be beyond range, or sample, or even logically impossible. • Other-var(s)=0 substantive meaning of 0 altered by rescaling • E.g., by “centering” (centering changes nothing, btw…) • Other-var(s)=0 may not have anything subst’ly main about it • COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE DERIVATIVES &/OR DIFFERENCES. • Only in purely linear-additive-separable model are they equal b/c only there do derivatives simply = coefficients. • SF is not “effect of SFrag ‘independent of’…” & definitely not its “effect ‘controling for’…other var(s) in the interax” • Cannot substitute linguistic invention for under-standing the model’s logic (i.e., its simple math)

15. Interpretation of Effects: NOTES2 • Interactions are logically symmetric: • For any function, not just lin-add. • If argue effect x depends z, must also believe effect z depends x. • Interactions often have 2nd-moment (variance, i.e., heteroskedacity) implications too: • Larger district magnitudes, DMag, are “permissive” elect sys: allow more parties… • Fewer Veto Actorsallow greater policy-change… (both need additional assumpts) • All of this holds for any type of variable: • Measurement: binary, continuous… • Level: micro or macro; i, j, k, …

16. Frequent 2nd-Moment Implications Interactions • DMag permissive ele sys: allows more parties… • Few Veto Actorsallows greater policy-change… • I.e., these are Rndm-Coeff &/or Het-sked Props…

17. Interpretation of Effects:Standard Errors for Effects • Std Errs reported with regression output are for coefficients, not for effects. • The s.e. (t-stat, p-level) for is std. err. for est’d effect SFrag at DMag=0 (…which is logically impossible). • Effect of x depends on z & v.v. (i.e., which was the point, remember?), so does the s.e.: • In words… More Generally:

18. From Hypotheses to Hypotheses Tests:Does Y Depend on X or Z?

19. From Hypotheses to Hypotheses Tests:Is Y’s Dependence on X Conditional on Z & v.v.? How? Does Y Depend on X, Z, or XZ?

20. Use & Abuse of Some Common ‘Rules’ • Centering to Redress Colinearity Concerns: • Adds no info, so changes nothing; no help with colinearity or anything else; only moves substantive content of x=0,z=0. • Specifically, makes coeff. on x (z), effect when z (x) at sample-mean, the new 0. Do only if aids presentation. • Must Include All Components (if x∙z, then x&z): • Application of Occam’s Razor &/or scientific caution (e.g., greater flexibility to allow linear w/in lin-interax model), but • Not a logical or statistical requirement. • Safer rule than opposite & to check almost always, but • Not override theory & evidence if (strongly) agree to exclude • Pet-Peeve: Lingistic Gymnastics to Dodge the Math • “Main effect, Interactive effect”: the effect in model is dy/dx. • Discussion of [coefficients & s.e.’s] as if [effects & s.e.’s].

21. Presentation: Marginal-Effects / Differences Tables & Graphs • Plot/Tabulate Effects, dy/dx, over Meaningful &/or Illuminating Ranges of z, with Conf. Int.’s • Explain axes • Explain shape • Linear-interax: • Will cross 0 & be insig @ 0. • Rescaling & • “main effect” • “centering” • Max(Asterisks)

22. Presentation: Expected-Value/Predictions Tables & Graphs • Predictions, E(y|x,z): • Here’s one place a little matrix algebra would help: • Use spreadsheet or stat-graph software (…list coming...)

23. Presentation:Choose Illuminating Graphics & Base Cases • Interpretation same regardless of “type” of interax: effect always ≡dy/dx, but present appropriately… • All combos Dummy, Discrete, or Continuous: • Dum-Dum=>4 (or 2#interax vars) points estimated, so box&whisk or histograms • Dum-Contin or DiscFew-Contin=>2 (or # categories) slopes, so E(y|x,z) as line or dy/dx as box&whisker or histograms • Contin(DiscMany)-Contin(DiscMany)=>Effect-lines best or (slices from) contour plot (i.e., slices from 3D) • Powers (e.g., X & X2=>parabola) viewable as interax w/ self; certain slope shifts too (e.g., dy/dx=a for x<x0 & b for x>x0 may see as x interacting w/ dummy for that condition) • Always plot over substantively revealing ranges. • Esp. w/ dums, have several (identical) specification options: • (full-set,set-1): choose which (& what base if use full-set-minus-1) to abet presentation & discussion • (overlap, disjoint): choose to facilitate presentation & discussion. • Scale Effectively: e.g., center only if & to extent that aids presentation & discussion (b/c centering does nothing else)

24. Examples & Practice:Basic Linear-Interactive Model • Basic Linear-Interactive Model: • Effect of edu? • For the record, the effect of lftrt: • Std Error of that Effect (of edu on eusupp)? • Std. Err. effect of edu: • To do this (e.g. in Stata, using the .dta subset) • First: gen edlr=education*leftright • Then: reg eu_supp ...education leftright edlr...

25. Examples & Practice:1aBasic Linear-Interactive Model:Education & Leftright Purely Micro-level Model • . reg eu_support education leftright edlr • Source | SS df MS Number of obs = 42680 • -------------+------------------------------ F( 3, 42676) = 371.33 • Model | 7869.51854 3 2623.17285 Prob > F = 0.0000 • Residual | 301474.935 42676 7.06427347 R-squared = 0.0254 • -------------+------------------------------ Adj R-squared = 0.0254 • Total | 309344.453 42679 7.24816545 Root MSE = 2.6579 • ------------------------------------------------------------------------------ • eu_support | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • education | .5918654 .0302344 19.58 0.000 .5326053 .6511255 • leftright | .080127 .0147551 5.43 0.000 .0512068 .1090472 • edlr | -.0320928 .0054447 -5.89 0.000 -.0427646 -.021421 • _cons | 5.093097 .0831665 61.24 0.000 4.930089 5.256105 • ------------------------------------------------------------------------------ • What’s effect of edu? Of lftrt? How moderate each other? • Only for modifying effect does standard regression output tell us directly. • What are the standard errors of these effects? • Only for modifying effect does standard regression output tell us directly. • Recall: Main Effects refer to beyond-range values; they not direct evidence on whether effect (generally) positive

26. Examples & Practice: Basic Linear-Interact Model1bEducation & Leftright Purely Micro-level Model • What are the standard errors of these effects? Need this: • .vce • Covariance matrix of coefficients of regress model • e(V) | education leftright edlr _cons • -------------+------------------------------------------------ • education | .00091412 • leftright | .00037584 .00021771 • edlr | -.00014825 -.00007456 .00002965 • _cons | -.00234221 -.00110293 .00037572 .00691666 • I prefer spreadsheet at this point: • Copy (“as Table”) regression-estimation results; Paste into spreadsheet. • Ditto for estimated v-cov of estimated coefficients (as text or table) • Finally, you’ll want summary stats for vars in your model (as text best): • . tabstat dgovpw psupgpw npgovpw PD NPPD NPGS NPPDGS, statistics( mean max min median iqr skewness sd ) columns(variables) • stats | dgovpw psupgpw npgovpw PD NPPD NPGS NPPDGS • ---------+---------------------------------------------------------------------- • mean | 25.24783 57.38261 1.965217 .6521739 1.369565 116.3004 765.9826 • max | 45.1 80.4 4.3 1 3.8 305.52 2181.2 • min | 11 41.1 1 0 0 49 0 • p50 | 26.6 57.2 1.8 1 1.6 96 916.2 • iqr | 13.7 10.8 1.7 1 2.2 83.47 1108 • skewness | .1587892 .821165 .907747 -.6390097 .2729543 1.218438 .1633625 • sd | 9.655468 9.103879 .9599284 .4869848 1.213723 69.04308 644.2565 • --------------------------------------------------------------------------------

27. Ex’s & Practice: Basic Linear-Interact Model1cEducation & Leftright Purely Micro-level Model • Spreadsht Formula to Plot Effect Lines w/ C.I. • Col 1: Conditioning Var (lftrt in deusup/dedu) • 1st Cell (A29): Enter min of range to consider (smpl min) • 2nd Cell: =A29+1 • Or sub “+(max-min)/(#steps)” for +1 (choose big# to smooth) • Copy down until reach max value you want plot to cover. • Column Two: Effect (dGovDur/dNP here) • 1st Cell (B29): Enter =$B$3+$B$5*$A29, where: • $B$3 is absolute reference to cell containing coefficient on edu • $B$5 is absolute reference to cell containing coefficient on edlr • $A29 is reference to cell containing value of lftrt for that row • $ optional, but helps if want copy whole block later for other effects • Copy Down

28. Ex’s & Practice: Basic Linear-Interact Model:Education & Leftright Purely Micro-level Model • Spreadsheet Formula to Plot Effect Lines w/ C.I. • Column Three: standard error (or can skip to bounds) • 1st Cell (A29):=($B$11+$D$13*$A29^2+2*$B$13*$A29)^0.5 • $B$11 is absolute reference to cell containing variance of edu coefficient • $D$13*$A29^2 is absolute reference to cell containing variance of edlr coefficient, times the square of the value of lftrt for that row. • 2*$B$13*$A29 is absolute reference to cell containing 2 times the covariance of edu and edlr coefficients times value of lftrt for that row. • ^0.5 turns that estimated variance into a standard error. • Copy down. • Column Four: Lower bound of 95% C.I. • 1st Cell (B29): Enter =+$B29-1.96*$C29, where: • $B29 is (column-absolute opt.) reference to cell containing effect est • $C29 is (column-absolute opt.) reference to cell containing std err est • 1.96 is critical value for 95% C.I. on large-N t-dist or std-norm dist; • can sub crit.val. for diff. C.I. % or smpl-size or use sprdsht formula • Copy Down • Column Five: Upper bound (analogous, but +1.96*$C29)

29. Ex’s & Practice: Basic Linear-Interact Model:Education & Leftright Purely Micro-level Model • In Stata, plot dY/dX w/ c.i. from smpl min-max: • egen zmin = min(z) ; egen zmax = max(z) finds those sample min & max for variable z. (z=psupgpw in our case, i.e., GS) • gen z0 = (_n-1)/(v-1)*(zmax-zmin)+zmin in 1/v creates var counting v equal-size steps from sample min to max. • gen dyhatdx=_b[x]+_b[xz]*z0creates var of dY/dX ests (x=npgovpw) • Stata code tedious to get to s.e.’s & c.i. plots (bit better in matrix form) • First have to work in matrices for bit, then back to vars: • matrix V = get(VCE) (makes matrix of v-cov mat) • matrix C= V[3,1] (grabs 3,1 element as covar) • gen column1 = 1 in 1/v (makes a variable equal to all ones) • mkmat column1, matrix(col1) (makes vector called col1 of that var) • matrix cov_x_xz = C*col1 (makes a constant vector of covar) • svmat cov_x_xz, name(cov_x_xz) (makes that vector a variable) • Finally, you can generate variances & std errors, which you could tabulate: • gen vardyhatdx=(_se[x])^2+(z0*z0)*(_se[xz]^2)+2*z0*cov_x_xz • gen sedyhatdx=sqrt(vardyhatdx) (makes variable equal to s.e. of effect) • tabdisp z0, cellvar(dyhatdx sedyhatdx) (makes table effects & s.e.’s) • Or you can generate the confidence interval bounds & plot: • gen LBdyhatdx=dyhatdx-invttail(e(df_r),.05)*sedyhatdx • gen UBdyhatdx=dyhatdx+invttail(e(df_r),.05)*sedyhatdx • twoway connected dyhatdx LBdyhatdx UBdyhatdx z0

30. Ex’s & Practice: Basic Linear-Interact Model Google ‘kam franzese michigan press’ for that data, and stata & excel resources or go directly to http://www.press.umich.edu /KamFranzese/Interactions.html Google ‘Matt Golder interaction’ or go directly to http://homepages.nyu.edu /~mrg217/interaction.html Stata help mfx Calculating Predicted-Values & Standard Errors

31. Elaborations, Complications, & Extensions:Sample-Splitting v. (Dummy-)Interacting • Split-sample (e.g., unit-by-unit) ≈ Full-Dummy Interax: • Subsample by binary (or multinomial, e.g., CTRY in TSCS) category to est sep’ly ≈ Dummy @ category & interact @ dummy w/ @ x. • Coeff’s same (or equal substantive content if using set-1 dummies). • S.E.’s same except s2 part of OLS’s s2(X'X)-1 is si2 for splitting • Can make exact by allow si2 (FWLS) • Subsample by hi/lo values some non-nominal var equivalent to nominalizing the extra-nominal info & dummy-interact; • I.e., wasting information, when usually have too little (non-parametric or extreme-measurement-error arguments might justify…) • Split-sample abets eyeballing, obfuscates statistical analysis, of the main point: the different effects by category. • What’s s.e/signif. of b1i-b1j? Need • Luckily, cov=0, but, still, squaring 2 terms, sum, & square-root in head? • Can choose full dummy set to mirror the split-sample estimates directly (& report that way, if wish) or the set-less-one to get significance of differences b/w samples directly (in the standard reported t-test) • One big advantage of hierarchical modeling is how it affords, naturally, various positions b/w these extremes

32. Cross-Level Interactions:From the (C or G)LRM to the HLM • If (C or G)LRM assumptions apply, then (O or G)LS unbiased, consistent, and efficient. • I.e., not much changes; +/- as before re: effects, v(effects), symmetry, neither micro-/macro-level coeff’s=effects, etc. • Two main issues of concern: • Parameter heterogeneity: (see pictures) • systematic &/or stochastic (fixed v. rndm intrcpt/coeff) • can cause bias if pattern unmodeled hetero relates to X, • Non-spherical error var-covar matrix (v-cov() will not be spherical): efficiency & proper s.e.’s issue, not a bias/consistency issue. • But “mere inefficiency” can be serious, & accurate s.e.’s very important.

33. From the (C or G)LRM to HLM • Examples of parameter hetereogeneity that covaries w/ X values, and so LS is biased: • Note re: FE v. RE -- both theoretically could cause bias if cov w/ X, but RE i.d.’d off orthogonality.

34. Elaborations, Complications, & Extensions:Random-Effect & Hierarchical/Multi-level Models • R.E. Model: Odd that std. lin-interax model: • Assumes know y=f(X)+error: • But dy/dx=f(z) w/o error!: • So, try: • => std. lin-interact again!...Except compound error-term… • HLM: Same model, except xij & zj,& • So it just std. lin-interact too, but w/ different compound-error stochastic properties…

35. Typical 2nd-Moment Implications of Interactions • DMag permissive ele sys: allow more parties… • Few Veto Actorsallow greater policy-change… • I.e., these are Rndm-Coeff &/or Het-sked Props…

36. From CLRM to Hierarchical Model:An Example • Std. HLM: Same model, except xij & zj,& different compound-error stochastic properties.

37. Properties of OLS under HLM Conditions • Properties of OLS Estimates of Lin-Interact Model if truly RE/HLM: • So, OLS coeff. est’s sill differ from truth by Aε*: • So, OLS coeff. est’s unbiased & consistent (iff…): • Note: only works for models with additively separable stochastic component; not nec’ly others (e.g., logit/probit)

38. Properties of OLS under HLM Conditions • But, OLS s.e.’s will be wrong; not s2(X'X)-1, but:

39. Sandwich Estimators • Not σ2I (even if each ε* is), so whole thing doesn’t reduce to σ2(X'X)-1, so OLS s.e.’s wrong. • Be OK on avg (unbiased) & in limit (consistent) if that term varied in way “ orthogonal to xx'” • But def’ly not b/c [∙] includes x & z, which part of X! • =brilliant insight of ‘robust’ (i.e., consistent) s.e. est’s: • Only need s.e. formula that accounts relation V(ε*) to “X'X”, i.e., regressors, squares, & cross-prod’s involved in X'[∙]X” •  “, robust” & “, cluster” can work (for RE & HLM, resp’ly) • so track e2 rel xx' & zz'heterosked.: • sim+grpngcluster:

40. From the CLRM to HLM •  appropriate “, robust” & “, cluster” can work • I.e., asymptotically s.e.’s right…BUT need large nj & N-k • Specifically, from small-sample corrections, seems need small: • I.e., coefficients still inefficient. • Want/need efficiency, or nj or N-k low? HLM/RE or FGLS/FWLS. • Note: similarity RE and HLM, RE & FWLS. As that sim suggests, RE only helps efficiency and only rightly does so if that’s all it does. (I.e., if the RE’s orthogonal to X.) • I.e., “work” thusly only or models with additively-separable stochastic components • As w/ all such “sandwich” estimators, a sort of logical disconnect in applying them to models w/o such separability…

41. Elaborations, Complications, & Extensions:Interax in QualDep (Inherently Interactive) Models • Probit/Logit Models w/ Interactions • Probit: – Logit: • Marginal Effects: (nonlinear, so must specify @ what x) • Start w/ x' pure lin-add, model inherently inter. b/c S-shaped: • Probit: • Logit: • If x'=…+xx+zz+xzxz… same except dx'/dx=xx+xzz; underlying propensity, i.e., movement along S-shape also interact explicitly x & z. [Discuss meaning inherent v. explicit interax…] • Probit: • Logit:

42. Elaborations, Complications, & Extensions:Interax in Nonlin/Qual (Inherently Interax) Models • Standard Errors? • Delta Method: • Probit Marginal-Effect s.e.: • Logit: same, except replaces • For first-difference effects, similar, but need specify from what x to what x, and not just at what x. • Or you could CLARIFY… or mfx…

43. Complex Context-Conditionality and Nonlinear Least-Squares • Complex Context Conditionality: The effect of anything depends on most everything else. E.g.: • Policymaking: • Socioeconomic-structure of interests • Party-system and internal party-structures • Electoral system & Governmental system • Socio-economic realities linking policies to outcomes • Voting: • Voter preferences & informational environment • Party/candidate locations & informational environment • Electoral & governmental system • Institutions: Sets of institutions; effect each depends configuration others present (e.g., that core of VoC claim). • Strategic Interdependence: each actors’ action depends on everyone else’s; complex feedback (see Franzese & Hays….)

44. Complex Context-Conditionality • EmpiricallyMulticolinear Nightmare: Options? • Ignore context conditionality (stay linear-additive): • Inefficient at best, biased more usually, and, anyway, context-conditionality is our interest! • Isolate one or some very few interactions for close study; ignore rest (stay linear-interactive): • Same, to degree lessened by amount interax allow, but demands on data rise rapidly w/ that amount. • “Structured Case Analysis”: • May help ‘theory generation’, but, for empirical evaluation, doesn’t help; worsens problem! (See Franzese OxfHndbk CP 2007). • EMTITM: Lean harder on thry/subst to specify more precisely the nature interax: functional form, precise measures, etc. • Refines question put to the data (changes default tests also). • GIVEN thry/subst. specification into empirical model, can estimate complex interactivity. Side benefits. But must give.

45. Nonlinear Least-Squares • Estimate NLS: • NLS is BLUE under same conditions OLS, w/  for X. • Interpreting NLS (already know how): Effects = deriv’s & 1st-diff’s; s.e.’s by Delta Method or simulation…

46. Generalized Nonlinear Least-Squares • GNLS: • GNLS is BLUE in same cond’s NLS, but  for I. • …don’t know , so need consistent 1st stage (e.g., NLS) • FGNLS is asymptotically BLUE:

47. Nonlinear Least-Squares & EMTI • EITM: Empirical Implications of Theoretical Models • Vision: Theory  more, sharper predictions  better tests, which therefore inform theory more, which… • TMEI: Theory-specified Models for Empirical Inference • Vision: Theory structures empirical models & relations b/w obs  specification & (causal) i.d. of empirical models • TIEM: Theoretical Implications of Empirical Measures • Vision: Emp. regularities, findings, measures inform theory dev’p. • EMTI: Empirical Models of Theoretical Intuitions • Vision: Intuitions derived from theoretical models specify empirical models. I.e., empirical specification to match intuitions, not model. • Note: Strongly counter some alternative moves stats & econometrics, & related; there toward non-parametric, matching, & experimentation—there, “model-dependence” a 4-letter word. Alternative audiences & rhetorical purposes? • Convince skeptic some causal effect exists, vs. • For the convinced, give richer, portable model of how world works.

48. Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model(Franzese, PA ‘03) • Monetary Policy in Open & Institutionalized Econ • Key C&IPE Insts/Struct: CBI, ER-Regime, Mon. Open • º CBI ≡ ºGovt Delegated Mon Pol to CB • º Peg ≡ ºDomestic (CB&Gov) Delegate to Peg-Curr (CB&Gov) • º FinOp ≡ ºDom cannot delegate b/c effectively del’d to globe • Effect of ev’thing to which for. & dom. mon. pol-mkrs would respond diff’ly depends on combo insts-structs & v.v., &, through intl inst-structs, for. on dom. & v.v. • Multicolinear Nightmare: • 23=8 inst-struct conds, i, times k factors per πi(Xi) if lin-interact • Exponentially more if all polynominials; k!/2(k-2)! if all pairs. • Good thing can lean on some thry to specify more precisely!

49. Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model • CB & Govt Interaction (Franzese, AJPS ‘99): • Full Monetary Exposure & Atomistic  zero domestic autonomy  • s.t. that, full e.r.fixCB&Gov match peg

50. Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model • Compact & intuitive, yet gives all theoretically expected interactions, in the form expected