Uniform Inference for Conditional Factor Models with Instrumental and Idiosyncratic Betas

Transcription

1 Uniform Inference for Conditional Factor Models with Instrumental and Idiosyncratic Betas Yuan Xiye Yang Rutgers University Dec 27 Greater NY Econometrics

2 Overview main results

3 Introduction Consider a factor model Y lt = β lt ft + u lt, l p, t T. Large p, large T. β l depends on observed instruments X: firm specific characteristics (e.g., Conner et al 5) size, value, momentum, volatility common macroeconomic variables (e.g., Ferson and Harvey 99 ) bonds spread, moody ratings

4 Introduction We propose: β lt = g lt (X lt ) + γ lt g part: captures long-run patterns in beta. γ part: captures high-frequency remaining components in beta X lt = (z l, x t, w lt ). z l : firm specific, time-invariant x t: common to stocks, e.g., macroeconomic variables w lt : both firm spec. and time-varying. This paper: Inference about instruments effects Effect of X on β represents instruments effects on risks.

5 Introduction y lt = β ltf t + u lt, β lt = g lt (X lt ) + γ lt Main challenge: DO NOT know whether γ lt is close to zero or not. Inference about g is uniformly valid over a class of DGP for γ Two ways of dealing of time-varying models. time-domain kernel smoothing, e.g., Ang et al. (2 ) fix time r, min ( ) t r (y lt β lt Tp f t ) 2 K Th l,t 2. High frequency data (this paper).

6 Main results. The distribution of ĝ has a discontinuity when var(γ it ). rate = var(γ + it ) pt p 2. Standard plug-in for asym. variance is only pointwise, NOT uniformly valid over the strength of var(γ it ). 3. cross-sectional bootstrap achieves uniformity. 4. When factors unknown, ĝ is biased, due to the effect of estimating factors. 5. Forecast using estimated factors: prediction interval is also misleading if time-varying γ it is ignored.

7 The econometric problem: A Toy example y lt = [x l θ + γ l ] f t + u lt matrix form: Y = [G + Γ]F + U, G = Xθ Key assumption: E(γ l x l ) =, E(u lt x l ) =. Estimations: Ĝ = P X YF(F F) ĝ l Two understandings: (i) first time series reg, then cross-sec reg. (ii) first cross-sec reg, then time series reg. recommend the second understanding (will get to this later)

8 The discontinuity issue Overview main results Fix l p, ĝ l g l = T p f th l (X m)u mt + p γ pt p m h l (X m) t= m= m= }{{}}{{} A A 2 A var(γ, A 2 m ) Tp p Case I: If var(γ m ), p(ĝl g l ) = pa 2 + o P () Case II: If var(γ m ), pt (ĝl g l ) = pt A + o P ()

9 Uniform Confidence interval Overview main results asym. var. = pt var( pt A ) + p var( pa 2 ) Usual plug-in method: use But this method is only pointwise. Goal: CI τ, pt var( pt A ) + p var( pa 2 ) sup P (g l CI τ ) ( τ). P P Here P is a broad class of DGP, containing various strenghs of γ l.

10 Poinwise Confidence interval var( pa 2 ) = p h m(x mt) 2 var(γ p mt X mt) m= var( pa 2 ) var( pa 2 ) equals p hm 2 [ γmt γ mt p γ mt mt] γ + p m= }{{} γ-estimation error p m= The γ-est. error is lower bounded by O P (T ). Hence when var( pa 2 ), hm 2 [ γmt γ mt Eγ mt mt] γ } {{ } LLN error var( pa 2 ) var( pa 2 ) T NOT negligible leads to over-coverage. If ignore var( pa 2 ), then when it is large, leads to under-coverage.

11 A possible solution The optimal rate: var( pa 2 ) var( pa 2 ) T To make optimal rate even faster : regularizations Use shrinkage, penalizations, thresholding, trimming... var( pa 2 ){ var( pa 2 ) k n} Similar idea used for distribution with discontinuity: finding binding moment inequalities inference for parameters on boundaries Random coefficients

12 Cross-sectional bootstrap Overview main results key assumption: cross-sectional independence Resample individuals, keep the entire time series for each sampled indiv. ĝ l g l = T p f th l (X m)u mt + p γ pt p m h l (X m) t= m= m= }{{}}{{} A A 2 ĝ l g l = pt Bootstrap var ( pa 2 ) = p p f thl (X m)umt + p m= m h2 mγ mt γ mt, so only has T t= m= p h 2 [ m γmt γ mt Eγ p mt γ mt] m= }{{} LLN error avoids γ-estimation error. cross-sec. bootstrap leads to uniformly valid C.I. p γ m h l (X m)

13 Why Cross-sectional bootstrap works? Key differences from Andrews (999, 2): Different sources of discontinuity in the asymptotic distribution : s f,t p n i F n i U mh i,ml, k np n i I t n m= and p γ p m,t h t,ml, m= }{{} (a) The discontinuity is a consequence of the interplay between the two terms. But the term (a) itself is continuous. The γ-estimation error can be avoided by resampling the cross-sectional units. Whether γ lt is near the boundary is unknown, and is uncertain under both the null and the alternative.

14 Formal treatments: High Frequency Setup Sample period [, T ], T <. Sampling interval length n. Number of times on each interval: k n, On each interval: n i Y = Y i n Y (i ) n = (G i + Γ i ) n i F + n i U + neglibigle terms, (.) Time variational error: Burkholder-Davis-Gundy Inequality ( E sup u [,k n n] Semimartingale increment over [t, t + u] ) = O( k n n)

15 Known factor case Ĝ t = i I n t Γ t = i I n t P i n i Y n i F i I n t (I p P i ) n i Y n i F n i F n i F i I n t P i : projection matrix using sieve basis of X i., n i F n i F, (.2) ĝ lt g lt = (A + A 2 ) + small

16 Unknown factor case: Replace n i F by n i F. How do we estimate n i F? Usual methods: PCA on We use: PCA on P n Y is noise-free : Then The errors vanish as p. ( n Y) ( n Y), T T (P n Y) (P n Y), T T P n Y = PG n F + PΓ n F + P n U + drift }{{} (P n Y) (P n Y) = n F G PG n F + errors n i F is also a GMM-type estimator.

17 Unknown factor case ĝ lt H ng lt bias = H n(a + A 2 ) + small bias: effect of estimating factors. Case I: If var(γ lt ), p(ĝlt H ng lt ) = ph na 2 + o P () Case II: If var(γ lt ), pkn(ĝ lt H ng lt bias) = pk nh na + o P () where var( n i U) is p p. bias = bias(var( n i U)) So bias-correction requires estimating a high-dim. cov. matrix.

18 Bootstrap confidence interval Fixed l p. Estimate ĝ lt, (and { n i F} i I n t if factors unknown.) 2. Take cross-sectional bootstrap { n i Y,..., n i Y p } i I n t and {X i,..., X ip} i I n t Do NOT bootstrap factors. 3. Estimate ĝ lt. 4. Repeat 2-3 B times, obtain ĝ lt,..., ĝ B lt. 5. Let q τ be the quantile of ĝ b lt ĝ lt CI τ = [ĝ lt ± q τ ] known factors CI τ = [ĝ lt bias ± q τ ] unknown factors

19 Discussion of E(γ X) = Overview main results Suppose and β l were known. β l = g(x l ) + γ l If E(γ l X l ), then need IV E(γ l w l ) =. Define Then T : g E(g(x l ) w l ) T g = E(β l w l ) g is identified iff x l w l is complete (Newey and Powell 3). Even if g is ident., T is discont. ill-posed problem. (Hall and Horowitz 6, Chen and Pouzo 2).

20 Simulations set Γ = w γn(, ), with various w γ. Construct CI using three methods: (i) the bootstrap confidence interval: ĝ ± q τ,bootstrap (ii) the over-coverage confidence interval: ĝ ±.96 var(a ) + var(a 2 ) use plug-in estimated variance for γ. (iii) the under-coverage confidence interval: ĝ ±.96 var(a ) ignore γ.

21 Simulations Table: Coverage probabilities, nominal probability= 95% w γ = strength of γ k n p bootstrap over under bootstrap over under Bootstrap is good uniformly over Γ. 2. The over is conservative when γ is weak, and becomes better as γ is stronger. 3. The under has under coverages as γ is stronger. 4. Bootstrap is good even if k n is small.

22 Out of sample forecast Overview main results We forecast y t+h = b y t + ρ F t + v t, t =,..., D e.g., y t = int. volatility = tt (t )T Each interval [(t )T, tt ] represents one day. Goal: prediction interval for σ 2 s ds y D+h D = b y D + ρ F D using estimated factors. Forecast is low frequency discrete time;

23 Estimating factors: high-frequency data n i Y = drift + [G i + Γ i ] n i F + n i U F t = tt (t )T df s Step : Estimate n i F De-mean (to remove drifts) F t = i Day t n i F F t = F t D D F t t= Step 2: Run OLS on y t+h = y t + ρ Ft + error Key assumptions: G is time-invariant. But Γ is still time varying.

24 Estimating factors y t+h = y t + ρ F t + v t, t =,..., D ŷ D+h D y D+h D = A A 4 + small A = D D t= wtvt A 2 = M D D t= Ft A 3 = p p i= U idh i A 4 = p DT t=(d )T n t F Γ t + D h D h t= A 4 = if Γ t is time-invariant, but not so in general. w d td s=(t )T n sf Γ s H So forecast interval is misleading if either (i) ignore Γ t, or (ii) treat Γ t time-invariant.

25 Forecast intervals Using this expansion, var(ŷ D+h D y D+h D ) = D var( DA ) + D var( DA 2 ) + p var( pa 3 ) + p var( pa 4 ) Plug-in estimated var works in this case. ( ) sup P yd+h D ŷ D+h D ± z τ/2 ŝ n ( τ). P P

26 Data Time period: 26 to 23. Sampling frequency: 5-min. 38 S&P5 component stocks. Firm characteristics: size, book to market ratio, momentum and volatility. Known factors: Market, HML, SMB and RMW.

27 Cross-sectional means of estimated G cross sectional mean Instrumental beta for Mkt Time small size medium size large size cross sectional mean Instrumental beta for HML Time small size medium size large size cross sectional mean Instrumental beta for SMB Time small size medium size large size cross sectional mean Instrumental beta for RMW small size - medium size large size Time Figure: Grouped by size. Size has negative effect on the betas for SMB risk factor.

28 Cross-sectional means of estimated G.5 Instrumental beta for Mkt 2 Instrumental beta for HML Instrumental beta.5 small vol medium vol large vol Instrumental beta - -2 small vol medium vol large vol Time Time Instrumental beta Instrumental beta for SMB small vol medium vol large vol Time Instrumental beta Instrumental beta for RMW small vol medium vol large vol Time Figure: Grouped by volatility. Volatility has positive effect on the betas for market factor.

29 Table: cross-sectional and time-series standard deviations of G and Γ Mkt HML SMB RMW size G Γ G Γ G Γ G Γ Averaged (over time) cross-sectional std small medium large Averaged (over firms) times-series std small medium large G has much smaller standard deviations than Γ in both cross-section and time series

30 2.5 2 Jul Jul Jul Jul Market vs Size HML vs Size SMB vs Size RMW vs Size

31 Table: Cross-sectional Proportion of significant G, 26 positive significance negative significance size Mkt HML SMB RMW Mkt HML SMB RMW small large Table: Cross-sectional Proportion of signs for β, Γ, 26 positive significance negative significance size Mkt HML SMB RMW Mkt HML SMB RMW β small.7.28 large Γ small.5.52 large.48.52

32 Other directions test relevance of instruments: g = x T θ + γ. H : θ i =. Confidence interval should be uniform in γ. Improve Fama-Macbeth regression using g in place of β. Inference about long-run instrumental effects: T gsds Test γ =