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The Anatolian hypothesis proposes that the dispersal of Proto-Indo-Europeans originated in Neolithic Anatolia. The hypothesis suggests that the speakers of the Proto-Indo-European language (PIE) lived in Anatolia during the Neolithic era, and associates the distribution of historical Indo-European languages with the expansion during the Neolithic revolution during the seventh and sixth millennia BC. The alternative and more academically favored view is the Kurgan hypothesis.
The main proponent of the Anatolian hypothesis was Colin Renfrew, who in 1987 suggested a peaceful Indo-Europeanization of Europe from Anatolia from around 7000 BC with the advance of farming by demic diffusion ("wave of advance"). Accordingly, most of the inhabitants of Neolithic Europe would have spoken Indo-European languages, and later migrations would at best have replaced these Indo-European varieties with other Indo-European varieties.
The main strength of the farming hypothesis lies in its linking of the spread of Indo-European languages with an archaeologically known event (the spread of farming) that is often assumed as involving significant population shifts.
A 2003 analysis of "87 languages with 2,449 lexical items" found an age range for the "initial Indo-European divergence" of 7,800–9,800 years, which was found to be consistent with the Anatolian hypothesis.  In 2006, the authors of the paper responded to their critics.  In 2011, the same authors and S. Greenhill found that two different datasets were also consistent with their theory. An analysis by Ryder and Nicholls (2011) found that: "Our main result is a unimodal posterior distribution for the age of Proto-Indo-European centred at 8400 years before Present with 95% highest posterior density interval equal to 7100–9800 years before Present." which was found to support the Anatolian Hypothesis.  A computerized phylogeographic study published in 2012 in Science, using methods drawn from the modeling of the spatial diffusion of infectious diseases, also supported the Anatolian hypothesis.
In a post on his popular blog, Greek Anthropologist Dienekes Pontikos wrote that he considered an origin of Indo-European in Neolithic Anatolia was "still the best hypothesis", citing the above studies as well as evidence that the methodology of the studies was correct. 
An argument against the Anatolian hypothesis is that PIE contains words for technologies that make their first appearance in the archaeological record in the Late Neolithic, in some cases bordering on the early Bronze Age, and that some of these words belong to the oldest layers of PIE. The lexicon includes words relating to agriculture (7500 BCE), metallurgy (7500 BCE), stockbreeding (6500 BCE) the plow (4500 BCE), gold (4500 BCE), domesticated horses (4000–3500 BCE) and wheeled vehicles (4000–3400 BCE). Horse breeding is thought to have originated with the Sredny Stog culture, semi-nomadic pastoralist living in the forest-steppe zone in what is now Ukraine. Wheeled vehicles are thought to have originated with Funnelbeaker culture in what is now Poland, Belarus, and parts of Ukraine.
Many Indo-European languages have cognate words meaning axle, for example, Latin axis, Lithuanian ašis, Russian os' , and Sanskrit ákṣa. (In some, a similar root is used for the word armpit: eaxl in Old English, axilla in Latin, and kaksa in Sanskrit.) All these are linked to the PIE root ak's-. The reconstructed PIE root i̯eu-g- gives rise to German joch, Hittite iukan, and Sanskrit yugá(m), all meaning yoke. Words wheel and cart/wagon/chariot take one of two common forms, thought to be linked with two PIE roots: the root kvel- becomes hvel (wheel) in Old Icelandic, kolo (wheel, circle) in Old Church Slavonic, kyklo (wheel, circle) in Greek, cakka/cakra (wheel) in Pali and Sanskrit, and kukäl (wagon, chariot) in Tocharian A. The root ret(h)- becomes rad (wheel) in Old High German, rota (wheel) in Latin, rãtas (wheel) in Lithuanian, and ratha (wagon, chariot) in Sanskrit.
Most estimates from Indo-Europeanists date PIE between 4500 and 2500 BC, with the most probable date falling right around 3700 BC. It is unlikely that late PIE (even after the separation of the Anatolian branch) post-dates 2500 BC, since Proto-Indo-Iranian is usually dated to just before 2000 BC. On the other hand, it is not very likely that early PIE predates 4500 BC, because the reconstructed vocabulary strongly suggests a culture of the terminal phase of the Neolithic bordering the early Bronze Age.
Reacting to criticism, Renfrew revised his proposal to the effect of taking a pronounced Indo-Hittite position. Renfrew's revised views place only Pre-Proto-Indo-European in 7th millennium BC Anatolia, proposing as the homeland of Proto-Indo-European proper the Balkans around 5000 BC, explicitly identified as the "Old European culture" proposed by Marija Gimbutas. He thus still situates the original source of the Indo-European language family in Anatolia around 7000BC. Reconstructions of a Bronze Age PIE society based on vocabulary items like "wheel" do not necessarily hold for the Anatolian branch, which appears to have separated from PIE at an early stage, prior to the invention of wheeled vehicles.
According to Renfrew (2004), the spread of Indo-European proceeded in the following steps:
Glottochronology has been used by some linguists to attempt to date languages by the changes in their lexicon over time. The method has been much criticized because of the assumption of a constant change rate. A revision of glottochronology by Sergei Starostin allows rate variation with time and dates the earliest PIE to 4670 BC, prior to the breakaway of Hittite, and the later PIE (origin of all other IE languages) to 3810 BC.