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- F7 (Stock market) vs. S&P 500 F8 (Money & credit) vs. M2SL F5 (Housing) vs. HOUST F6 (Labor) vs. PAYEMS F3 (Spreads) vs. T10YFFM F4 (Interest rates) vs. GS1 F1 (Output) vs. INDPRO F2 (Prices) vs. CPIAUCSL 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 −5.0 −2.5 0.0 2.5 5.0 −10 −5 0 5 −2.5 0.0 2.5 5.0 −5 0 5 −6 −4 −2 0 2 4 −4 −2 0 2 −2 0 2 −6 −4 −2 0 2 Estimated factor Observed variable Figure C7: Factors estimated with SPCA, compared with some representative observed variables (FRED-MD).
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- Here we drop the index 0 which Doz, Giannone, and Reichlin (2011) use for the true values of the parameters.
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- i) Doz, Giannone, and Reichlin (2011) have shown that (see Lemma 2 in their Appendix): 1 N p D D “ O ˆ 1 N ˙ ` OP ˆ 1 ? T ˙ We can thus write: p D “ D ` Op1q ` OP N ? T . As all the terms of D tend to infinity linearly with N, we thus obtain p D “ OP pNq.
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- ii) Doz, Giannone, and Reichlin (2011) have also shown in the same lemma that: N p D1 D1 “ O ˆ 1 N ˙ ` OP ˆ 1 ? T ˙ We can thus write: p D1 “ D1 ` O ` 1 N2 ˘ ` OP 1 N ? T . As all the terms of D tend to infinity linearly with N, we thus obtain p D1 “ OP ` 1 N ˘ .
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- iii) If we denote p Σe “ X1X T p Λp Λ1, Doz, Giannone, and Reichlin (2011) have shown that:2 p σij,e σij,e “ OP ˆ 1 ? N ˙ ` OP ˆ 1 ? T ˙ (A1) and that the result is uniform w.r.t. pi, jq. By Assumption (CR4) we know that σmin pΣeq “ c ą 0. By the Weyl theorem, we know that: σmin p Σe ě σmin p Σe Σe ` σmin pΣeq (A2) 2
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- Note. Shaded areas indicate periods of recession as defined by the National Bureau of Economic Research. The observed variables are stationarized using the transformations recommended by McCracken and Ng (2016). The correlations between the factors estimated with SPCA and the representative observed variables are respectively 0.96, 0.89, 0.95, 0.96, 0.99, 0.91, 0.94, 0.61.
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- To prove our theorem, we use properties which have been obtained by Doz, Giannone, and Reichlin (2011) under their set of assumptions or which can be easily derived from properties which they have stated. In particular, we use the following properties. Preliminary properties (PCA properties) i) p D “ OP pNq ii) p D1 “ OP ` 1 N ˘ iii) X1X T 1 “ OP p1q Proof.
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