Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction Optimal adaptive detection of small Plan of the talk Some testing correlation functions procedures Main goals Construction of the test Alain Guay Emmanuel Guerre Stepana Lazarova Null limit distribution Rate consistency Sparse alternatives Vienna, 26 July 2008 Simulations Applications Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 1 / 26
Introduction Motivation Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction Descriptive statistic: is f u t g is a (weak) white noise? Plan of the talk Model diagnostic Some testing � � procedures b = X t � b θ 0 � b θ 1 X t � 1 � � � � � b AR : b u t = u t θ θ k X t � k Main goals � � = X 2 b Construction of ARCH : b t u t = u t θ t � 1, where the test σ 2 b σ 2 t = b θ 0 + b θ 1 X 2 t � 1 + � � � + b θ k X 2 Null limit b � � t � k distribution plim b If f u t = u t θ g is not a white noise, the lag Rate consistency Sparse alternatives order k should be increased Simulations Economics : Market E¢ciency and Rational Applications Expectation Hypothesis Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 2 / 26
Introduction Framework Adaptive detection of small correlations Guay, Guerre, Lazarova f u t g stationary process with E [ u t ] = 0 and …nite Introduction variance R 0 Plan of the talk Some testing Covariance function: R j = cov ( u t , u t + j ) procedures Sample covariances: b R j = 1 n ∑ n Main goals t = j + 1 b u t b u t � j Construction of Hypotheses the test Null limit H 0 : R j = 0 for all j 6 = 0 distribution H A : R j 6 = 0 for some j 6 = 0 Rate consistency Sparse alternatives Technical conditions Simulations Absolute summability Cumulants of f u t g up to 8th Applications b u t ( θ ) twice di¤erentiable w.r.t to θ Final remarks θ n 1 / 2 consistent b Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 3 / 26
Plan of the talk Rest of the talk Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction Some testing procedures: Cramer von Mises tests and Plan of the talk smooth tests Some testing procedures Main goals of the talk Main goals The test Construction of Null limit distribution the test Null limit Rate consistency distribution Adaptive rate optimality: "sparse" alternatives Rate consistency (Simulations) Sparse alternatives Applications to …nancial squared returns Simulations Final remarks Applications Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 4 / 26
Some testing procedures Cramer von Mises type tests Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction b ! 2 Plan of the talk n � 1 CvM = n 1 R j ∑ , Some testing π 2 j 2 b procedures σ j j = 1 Main goals σ j = b f u t g observed, b R 0 or heteroscedasticity robust (Lobato Construction of the test et al, 2001) Null limit distribution Does not use any smoothing parameters Rate consistency Detects n 1 / 2 Pitman local alternatives Sparse alternatives Simulations Not suitable for alternatives with small correlations at Applications low order Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 5 / 26
Some testing procedures Smooth test statistics Adaptive detection of small correlations Guay, Guerre, Lazarova b ! 2 p R j Introduction c ∑ BP p = n (Box and Pierce 1970), b σ j Plan of the talk j = 1 Some testing � b ! 2 � j procedures n � 1 R j b ∑ K 2 S p = n (Hong 1996) Main goals b p σ j j = 1 Construction of the test p = truncation/smoothing parameter Null limit distribution Does not downweight large j , but p is di¢cult to Rate consistency choose in practice Sparse alternatives Asymptotically minimax (Ermakov, 1994) against Simulations smooth alternatives, but for a p dependent of the Applications Final remarks unknown alternative Does not detect n 1 / 2 Pitman local alternatives if p is too large Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 6 / 26
Some testing procedures Data-driven choice of the smoothing parameter Adaptive detection of small correlations Guay, Guerre, Lazarova Fan and Yao (2005) propose Introduction Plan of the talk b S p � E ( p ) Some testing max procedures V ( p ) p 2P Main goals Construction of No theoretical study the test Lack of proper critical values for usual sample sizes Null limit distribution “Rule of thumb” for smooth tests (Hong (1996), Rate consistency Andrews (1991), Newey West (1994)) Sparse alternatives are in general “optimal” for estimation of the spectral Simulations density but not for testing Applications No clear limit distribution under the null due to a Final remarks random choice of p Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 7 / 26
Main goals Main goals of the talk Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction Plan of the talk Some testing procedures To propose a test that Main goals has a simple null limit distribution 1 Construction of the test achieves adaptive optimal detection of "sparse" 2 Null limit distribution alternatives Rate consistency Sparse alternatives Simulations Applications Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 8 / 26
Construction of the test Construction of the test: improving a …rst initial Adaptive detection of small correlations test Guay, Guerre, Lazarova Many applied works use a moderate deterministic p Introduction (typically, 5, 10, or ln n ). The test rejects H 0 if Plan of the talk b Some testing S p � E ( p ) procedures � z n ( α ) , V ( p ) Main goals Construction of where � � K 2 � � the test E ( p ) = ∑ n � 1 1 � j j and Null limit j = 1 n p distribution � � 2 K 4 � � V 2 ( p ) = 2 ∑ n � 1 1 � j j Rate consistency are approximation j = 1 n p Sparse alternatives of the mean and variance of b S p under the null of Simulations independence (Hong, 1996) Applications E ( p ) + V ( p ) z n ( α ) is a critical value: Final remarks Null of indepence: Chi square (Box-Pierce, 1970) or Fisher, Normal z n ( α ) (Hong, 1996, p ! ∞ , estimated residuals) Null of non correlation: mixture of Chi squares or Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 9 / 26
Construction of the test When should I increase my favorite trunctation Adaptive detection of small correlations parameter? Guay, Guerre, Lazarova I should change my favorite p for a larger p if b S p Introduction strongly di¤ers from b S p , i.e. if for a level λ n typically Plan of the talk Some testing tending to 0 procedures ( b S p � b S p ) � E ( p , p ) Main goals � z n ( λ n ) , where Construction of V ( p , p ) the test Null limit distribution E ( p , p ) = E ( p ) � E ( p ) , � � 2 � � j � � j �� 2 Rate consistency n � 1 1 � j V 2 ( p , p ) K 2 � K 2 Sparse alternatives ∑ = 2 . n p p Simulations j = 1 Applications The …nal test uses the smoothing parameter �� � � Final remarks S p � b b e p = arg max S p � E ( p , p ) � z n ( λ n ) V ( p , p ) p 2 f p , p g Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 10 / 26
Construction of the test The proposed test Adaptive detection of small correlations Guay, Guerre, � � Lazarova p , 2 p , ..., 2 Q � 1 p = p γ n � 0 penalty sequence, P = Introduction �� � � Plan of the talk S p � b b b p = arg max S p � E ( p , p ) � γ n V ( p , p ) . Some testing p 2P procedures Main goals Rejects H 0 if Construction of the test b S b p � E ( p ) Null limit � z n ( α ) . distribution V ( p ) Rate consistency � � ! 1 under the null: Sparse alternatives The test uses that P b p = p Simulations for the studentization with E ( p ) and V ( p ) (improves Applications power) Final remarks � � when choosing the critical value z n ( α ) = z n α ; p Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 11 / 26
Null limit distribution A penalty lower bound Adaptive detection of small correlations Guay, Guerre, Lazarova Introduction Plan of the talk Theorem Some testing Suppose that u t is identically distributed. Then, if procedures p = o ( n 1 / 3 ) , and if the selection sequence f γ n , n � 1 g Main goals Construction of satis…es the test Null limit γ n � ( 2 ln ln n ) 1 / 2 + ǫ for some ǫ > 0 , (1) distribution Rate consistency b p = p with a probability tending to 1 under the null, and the Sparse alternatives test is asymptotically of level α . Simulations Applications Final remarks Guay, Guerre, Lazarova (UQAM, QM) Adaptive detection of small correlations Vienna, 26 July 2008 12 / 26
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